Dispersal and Good Habitat Quality Promote Neutral Genetic Diversity in Metapopulations

Abstract

Dispersal is a fundamental and crucial ecological process for a metapopulation to survive in heterogeneous or changing habitats. In this paper, we investigate the effect of the habitat quality and the dispersal on the neutral genetics diversity of a metapopulation. We model the metapopulation dynamics on heterogeneous habitats using a deterministic system of ordinary differential equations. We decompose the metapopulation into several neutral genetic fractions seeing as they could be located in different habitats. By using a mathematical model which describes their temporal dynamics inside the metapopulation, we provide the analytical results of their transient dynamics, as well as their asymptotic proportion in the different habitats. The diversity indices show how the genetic diversity at a global metapopulation scale is preserved by the correlation of two factors: the dispersal of the population, as well as the existence of adequate and sufficiently large habitats. The diversity indices show how the genetic diversity at a global metapopulation scale is preserved by the correlation of two factors: the dispersal of the population as well as the existence of adequate and sufficiently large habitats. Moreover, they ensure genetic diversity at the local habitat scale. In a source–sink metapopulation, we demonstrate that the diversity of the sink can be rescued if the condition of the sink is not too deteriorated and the migration from the source is larger than the migration from the sink. Furthermore, our study provides an analytical insight into the dynamics of the solutions of the systems of ordinary differential equations.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

References

  1. Bansaye V, Méléard S (2015) Stochastic models for structured populations. Springer, Berlin

    Google Scholar 

  2. Barton NH (2000) Genetic hitchhiking. Philos Trans R Soc Lond B Biol Sci 355(1403):1553–1562. https://doi.org/10.1098/rstb.2000.0716

    Article  Google Scholar 

  3. Bohonak AJ (1999) Dispersal, gene flow, and population structure. Q Rev Biol 74(1):21–45. https://doi.org/10.1086/392950

    Article  Google Scholar 

  4. Bohrer G, Nathan R, Volis S (2005) Effects of long-distance dispersal for metapopulation survival and genetic structure at ecological time and spatial scales. J Ecol 93(5):1029–1040

    Article  Google Scholar 

  5. Bonnefon O, Coville J, Garnier J, Hamel F, Roques L (2014) The spatio-temporal dynamics of neutral genetic diversity. Ecol Complexity 20:282–292

    Article  Google Scholar 

  6. Charlesworth B, Morgan MT, Charlesworth D (1993) The effect of deleterious mutations on neutral molecular variation. Genetics 134(4):1289–1303

    Article  Google Scholar 

  7. Edwards AWF (1963) Migrational selection. Heredity 18:101–106

    Article  Google Scholar 

  8. Fournier N, Méléard S (2004) A microscopic probabilistic description of a locally regulated population and macroscopic approximations. Ann Appl Probab 14(4):1880–1919

    MathSciNet  Article  Google Scholar 

  9. Freckleton RP, Watkinson AR (1990) Large-scale spatial dynamics of plants: metapopulations, regional ensembles and patchy populations. J Ecol 90:419–434

    Article  Google Scholar 

  10. Garnier J, Lewis MA (2016) Expansion under climate change: the genetic consequences. Bull Math Biol 78(11):2165–2185. https://doi.org/10.1007/s11538-016-0213-x

    MathSciNet  Article  MATH  Google Scholar 

  11. Garnier J, Giletti T, Hamel F, Roques L (2012) Inside dynamics of pulled and pushed fronts. J Math Pures Appl 11:173–188

    MathSciNet  MATH  Google Scholar 

  12. Gyllenberg M, Sderbacka G, Ericsson S (1993) Does migration stabilize local population dynamics? Analysis of a discrete metapopulation model. Math Biosci 118(1):25–49. https://doi.org/10.1016/0025-5564(93)90032-6

    MathSciNet  Article  MATH  Google Scholar 

  13. Hallatschek O, Nelson DR (2008) Gene surfing in expanding populations. Theor Popul Biol 73:158–170

    Article  Google Scholar 

  14. Hewitt GM (2000) The genetic legacy of the quarternary ice ages. Nature 405:907–913

    Article  Google Scholar 

  15. Holt RD (1985) Population dynamics in two-patch environments: some anomalous consequences of an optimal habitat distribution. Theor Popul Biol 28:181–208

    MathSciNet  Article  Google Scholar 

  16. Husband BC, Barrett SCH (1996) A metapopulation perspective in plant population biology. J Ecol 84:461–469

    Article  Google Scholar 

  17. Ingvarsson P (1997) The effect of delayed population growth on the genetic differentiation of local populations subject to frequent extinctions and recolonizations. Evolution 51:29–35

    Article  Google Scholar 

  18. Iverson LR, Prasad A, Schwartz MW (1999) Modeling potential future individual tree-species distributions in the eastern united states under a climate change scenario: a case study with pinus virginiana. Ecol Model 115:77–93

    Article  Google Scholar 

  19. Jenouvrier S, Garnier J, Patout F, Desvillettes L (2017) Influence of dispersal processes on the global dynamics of emperor penguin, a species threatened by climate change. Biol Conserv 212:63–73. https://doi.org/10.1016/j.biocon.2017.05.017

    Article  Google Scholar 

  20. Lande R (1992) Neutral theory of quantitative genetic variance in an island model with local extinction and colonization. Evolution 46(2):381–389. https://doi.org/10.1111/j.1558-5646.1992.tb02046.x

    Article  Google Scholar 

  21. Levin SA, Paine RT (1974) Disturbance, patch formation, and community structure. Proc Natl Acad Sci 71(7):2744–2747. https://doi.org/10.1073/pnas.71.7.2744

    Article  MATH  Google Scholar 

  22. Levin SA, Muller-Landau HC, Nathan R, Chave J (2003) The ecology and evolution of seed dispersal: a theoretical perspective. Annu Rev Ecol Evol Syst 34:575–604

    Article  Google Scholar 

  23. Lynch M (1988) The divergence of neutral quantitative characters among partially isolated populations. Evolution 42(3):455–466

    Article  Google Scholar 

  24. Malanson GP, Cairns DM (1997) Effects of dispersal, population delays, and forest fragmentation on tree migration rates. Plant Ecol 131(1):67–79. https://doi.org/10.1023/A:1009770924942

    Article  Google Scholar 

  25. Maruyama T, Kimura M (1980) Genetic variability and effective population size when local extinction and recolonization of subpopulations are frequent. Proc Natl Acad Sci USA 77:6710–6714

    Article  Google Scholar 

  26. Moody M (1981) Polymorphism with selection and genotype-dependent migration. J Math Biol 11:245–267

    MathSciNet  Article  Google Scholar 

  27. Pannell JR, Charlesworth B (1999) Neutral genetic diversity in a metapopulation with recurrent local extinction and recolonization. Evolution 53(3):664–676. https://doi.org/10.1111/j.1558-5646.1999.tb05362.x

    Article  Google Scholar 

  28. Pannell JR, Charlesworth B (2000) Effects of metapopulation processes on measures of genetic diversity. Philos Trans R Soc B: Biol Sci 355(1404):1851–1864. https://doi.org/10.1098/rstb.2000.0740

    Article  Google Scholar 

  29. Parsons PA (1963) Migration as a factor in natural selection. Genetica 33(1):184–206. https://doi.org/10.1007/BF01725761

    Article  Google Scholar 

  30. Ponchon A, Garnier R, Grémillet D, Boulinier T (2015) Predicting population response to environmental change: the importance of considering informed dispersal strategies in spatially structured population models. Divers Distrib 21:88–100

    Article  Google Scholar 

  31. Roques L, Garnier J, Hamel F, Klein E (2012) Allee effect promotes diversity in traveling waves of colonization. Proc Natl Acad Sci USA 109:8828–8833

    MathSciNet  Article  Google Scholar 

  32. Roques L, Walker E, Franck P, Soubeyrand S, Klein EK (2016) Using genetic data to estimate diffusion rates in heterogeneous landscapes. J Math Biol 73(2):397–422. https://doi.org/10.1007/s00285-015-0954-4

    MathSciNet  Article  MATH  Google Scholar 

  33. Schwartz MW, Iverson LR, Prasad AM (2001) Predicting the potential future distribution of four tree species in ohio using current habitat availability and climatic forcing. Ecosystems 4:568–581

    Article  Google Scholar 

  34. Simpson EH (1949) Measurment of diversity. Nature 163:688

    Article  Google Scholar 

  35. Slatkin M (1977) Gene flow and genetic drift in a species subject to frequent local extinctions. Theor Popul Biol 12:253–262

    Article  Google Scholar 

  36. Slatkin M (1985) Gene flow in natural populations. Annu Rev Ecol Syst 16:393–430

    Article  Google Scholar 

  37. Slatkin M (1987) The average number of sites separating dna sequences drawn from a subdivided population. Theor Popul Biol 32:42–49

    Article  Google Scholar 

  38. Smith JM, Haigh J (1974) The hitch-hiking effect of a favorable gene. Genet Res 23(1):2335. https://doi.org/10.1017/S0016672300014634

    Article  Google Scholar 

  39. Souza FL, Cunha AF, Oliveira MA, Pereira GAG, Reis SFd (2002) Estimating dispersal and gene flow in the neotropical freshwater turtle hydromedusa maximiliani (chelidae) by combining ecological and genetic methods. Genet Mol Biol 25(2):151–155. https://doi.org/10.1590/S1415-47572002000200007

    Article  Google Scholar 

  40. Travis JMJ, Mustin K, Bartoń KA, Benton TG, Clobert J et al (2012) Modelling dispersal: an eco-evolutionary framework incorporating migration, movement, settlement behaviour and the multiple costs involved. Methods Ecol Evol 3(4):628–641

    Article  Google Scholar 

  41. van Heerwaarden J, van Eeuwijk FA, Ross-Ibarra J (2010) Genetic diversity in a crop metapopulation. Heredity 104(1):28–39. https://doi.org/10.1038/hdy.2009.110

  42. Visser ME (2008) Keeping up with a warming world; assessing the rate of adaptation to climate change. Proc R Soc B: Biol Sciences 275(1635):649–659. https://doi.org/10.1098/rspb.2007.0997

    Article  Google Scholar 

  43. Watts AG, Schlichting PE, Billerman SM, Jesmer BR, Micheletti S, Fortin MJ, Funk WC, Hapeman P, Muths E, Murphy MA (2015) How spatio-temporal habitat connectivity affects amphibian genetic structure. Front Genet 6: https://doi.org/10.3389/fgene.2015.00275

  44. Whitlock M, Barton N (1997) The effective size of a subdivided population. Genetics 146:427–441

    Article  Google Scholar 

  45. Wright S (1949) The genetical structure of populations. Ann Eugen 15(1):323–354. https://doi.org/10.1111/j.1469-1809.1949.tb02451.x

    MathSciNet  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Jimmy Garnier.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

JG acknowledges NONLOCAL project (ANR-14-CE25-0013), GLOBNETS project (ANR-16-CE02-0009) and the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation program (Grant Agreement No. 639638, MesoProbio).

Appendices

Appendix A: Properties of the Asymptotic Proportion \(p^*\)

In this section, we are interested in the qualitative properties of the asymptotic proportion of a fraction in a metapopulation at equilibrium. More precisely, we either consider a metapopulation in a spatially structured favorable environment or a source–sink metapopulation at equilibrium \(\mathbf {N}^*\). And we consider a fraction with density \(\mathbf {n}\) inside this metapopulation which satisfies equation (2). We know from our previous result that the fraction will converge to an asymptotic proportion of the metapopulation \(p^*\) defined by (7). We investigate the effects of the quality of the habitats, that is, its growth rate \(r_i\), and the migration rates \(\varepsilon _{ij}\) on the asymptotic proportion \(p^*\). These properties will give us insight into the behavior of the diversity which is explain in the main text.

A.1: Scenario 1: Metapopulation in Spatially Structured Favorable Environment

We consider a metapopulation over two good habitats at equilibrium. Using the notation of Theorem 1, the asymptotic proportion of the fraction with initial density \(\mathbf {n}(0)=(n_1(0),n_2(0))\) is

$$\begin{aligned} p^* = \dfrac{n_1(0)\varepsilon _{12}N_1^* + n_2(0)\varepsilon _{21}N^*_2}{ \varepsilon _{12}(N_1^*)^2 + \varepsilon _{21}(N^*_2)^2} . \end{aligned}$$
(10)

A.1.1: Effect of the Growth Rate on Asymptotic Proportion

We first look at the effect of the habitat quality defined by the growth rate of the habitat \(r_i\) on the asymptotic proportion \(p^*\).

Proposition 6

(Monotonicity of \(p^*\) with respect to \(r_2\)) For any initial condition \(\mathbf {n}(0)\), the asymptotic proportion \(p^*\) associated with the fraction \(\mathbf {n}\) solving (6) is monotonic with respect to the growth rate \(r_2\). More precisely, the function \(r_2\mapsto p^*(r_2)\) is

  • decreasing with respect to \(r_2\) if   \((\varepsilon _{21} K_2-\varepsilon _{12}K_1)\big (p_2(0)-p_1(0)\big )\le 0 ,\)

  • increasing with respect to \(r_2\) if  \((\varepsilon _{21} K_2-\varepsilon _{12}K_1)\big (p_2(0)-p_1(0)\big )\ge 0\) .

Our result shows that for a given fraction v , the impact of the growth rate of one habitat on the asymptotic contribution of this habitat mainly depends on the flux of individuals between the habitats. In particular, enhancing the quality of habitat 2 will promote contribution of habitat 2 if the flux of individuals from habitat 2 is higher than the one of habitat 1 (it corresponds to the case \(\varepsilon _{21} K_2\ge \varepsilon _{12}K_1\)). Conversely, if the mean individuals flux from habitat 2 is smaller than the one from habitat 1, then the contribution of a poor quality habitat 2 is better than the contribution of a good quality habitat.

Let us now look at the relative contribution of the two habitats. More precisely, we assume that the fraction v is only present in one habitat, that is either \(p_1(0)=1\) and \(p_2(0)=0\) or \(p_1(0)=0\) and \(p_2(0)=1\). In this case, we compare the asymptotic proportion \(p^*\) with respect to 1/2 which corresponds to an equal contribution from each habitat.

Proposition 7

(Contribution of each habitat) Let us consider a fraction that fully occupied only one habitat, either \(p_1(0)=1\) and \(p_2(0)=0\) or \(p_1(0)=0\) and \(p_2(0)=1\). Then, the asymptotic proportion satisfies the following conditions:

  • If \((\sqrt{\varepsilon _{21}} K_2-\sqrt{\varepsilon _{12}}K_1)(\varepsilon _{21}-\varepsilon _{12})\le 0\), then the sign of \((p^*(r_2)-1/2)\) does not depend on \(r_2\). In addition, \(p^*(r_2)\le 1/2\) if \(\big (p_2(0)-p_1(0)\big )(\varepsilon _{21}-\varepsilon _{12})>0\) and \(p^*(r_2)\ge 1/2\) if \(\big (p_2(0)-p_1(0)\big )(\varepsilon _{21}-\varepsilon _{12})<0\).

  • If \((\sqrt{\varepsilon _{21}} K_2-\sqrt{\varepsilon _{12}}K_1)(\varepsilon _{21}-\varepsilon _{12})> 0\), then there exists \(\bar{r}_2>0\) such that \(p^*(\bar{r}_2)=1/2\) where

    $$\begin{aligned} \bar{r}_2= \dfrac{\sqrt{\varepsilon _{21}}[\sqrt{\varepsilon _{21}} -\sqrt{\varepsilon _{12}}]}{1-\dfrac{\varepsilon _{12}K_1}{\varepsilon _{21}K_2} \sqrt{\dfrac{\varepsilon _{21}}{\varepsilon _{12}}}\left[ 1+ \dfrac{\sqrt{\varepsilon _{12}}[\sqrt{\varepsilon _{21}}-\sqrt{\varepsilon _{12}}]}{r_1}\right] } \end{aligned}$$
    (11)

    In addition, we get

    • \(p^*(r_2) \ge \dfrac{1}{2}\) if \((r_2 - \bar{r}_2)\big (p_2(0)-p_1(0)\big )(\varepsilon _{21}-\varepsilon _{12})>0\);

    • \(p^*(r_2) \le \dfrac{1}{2}\) if \((r_2 - \bar{r}_2)\big (p_2(0)-p_1(0)\big )(\varepsilon _{21}-\varepsilon _{12})<0\).

Our result shows that the relative contribution of each habitat may depend on the relative quality of the habitat (see Fig. 12). In particular, in an environment with a large habitat and a small habitat with smaller migration from the largest habitat than from the smallest habitat, the contribution of the large habitat is larger than the contribution of the smallest habitat whatever the quality of the habitats is.

Moreover, if we assume that habitat 1 is larger than habitat 2 (\(K_1>K_2\)), then our result shows that the relative quality of the habitats may have influence on the relative contribution if the flux of individuals is either really high or low. More precisely, if migration from habitat 1 is really smaller than the migration from habitat 2 ( \(\sqrt{\varepsilon _{21}}K_2>\sqrt{\varepsilon _{12}}K_1\) ), then its relative contribution is large only if the quality of habitat 2 is really poor (\(p^*(r_2)>1/2\) if \(r_2\le \bar{r}_2\)). Conversely, if the migration from habitat 1 is larger than migration from habitat 2 (\(\varepsilon _{12}>\varepsilon _{21}\)), its relative contribution is large as long as the quality of habitat 2 is good enough (\(p^*(r_2)>1/2\) if \(r_2\ge \bar{r}_2\)). However, when the fluxes between the habitats are similar (\(\sqrt{\varepsilon _{21}}K_2<\sqrt{\varepsilon _{12}}K_1\) and \(\varepsilon _{12}<\varepsilon _{21}\) ), then the contribution of habitat 1 is always larger than the one of habitat 2 whatever the quality of the habitats.

Fig. 12
figure12

(Color figure online) Proportion of the subgroup of gene 1 in the case of scenario 1; in figure a first habitat: carrying capacity \( K_{1}=100\)); growth rate \(r_1 =1 \), migration rate from habitat1 to habitat2 \(\varepsilon _{12}=0.1\); second habitat: carrying capacity \( K_{2}=1000\), migration rate from habitat2 to habitat1 \(\varepsilon _{21}\in \left\{ 0.01 \, , 0.1 \, ,0.5 \, ,0.7 \, , 1\right\} . \) In figure b first habitat: carrying capacity \( K_{1}=500\)); growth rate \(r_1 =1 \), migration rate from habitat1 to habitat2 \(\varepsilon _{12}=0.7\); second habitat: carrying capacity \( K_{2}=300\), migration rate from habitat2 to habitat1 \(\varepsilon _{21}\in \left\{ 0.01 \, , 0.1 \, ,0.5 \, ,0.7 \, , 1\right\} \)

Proof of Proposition 6

Let \(\varepsilon _{12}\) and \(\varepsilon _{21}\) be in (0, 1], \(\mathbf {F}\) satisfy hypothesis of Proposition 10, \(p_1(0)\) and \(p_2(0)\) be in [0, 1] and \(\mathbf {N}^*=(N_1^*,N^*_2)\) be a stationary state of  (1). We begin to study the variation of \(p^*\) with respect to \(r_2\). Using Theorem 1, we have:

$$\begin{aligned} p^*(r_2) = \dfrac{p_1(0)\varepsilon _{12} + p_2(0)\varepsilon _{21}(P^*)^2}{\varepsilon _{12} + \varepsilon _{21}(P^*)^2} \ \hbox { where } \ P^* = \dfrac{N^*_2}{N^*_1} . \end{aligned}$$

Differentiating the expression with respect to \(r_2\), we obtain:

$$\begin{aligned} \partial _{r_2} p^*(r_2) = 2\dfrac{(p_2(0)-p_1(0))\varepsilon _{12}\varepsilon _{21}}{(\varepsilon _{12} + \varepsilon _{21}(P^*)^2)^2}P^*\partial _{r_2} P^*. \end{aligned}$$

The variations of \(p^*(r_2)\) only depend on the sign of \((p_2(0)-p_1(0))\partial _{r_2} P^*\). From the definition of \(P^*\), we have

$$\begin{aligned} \partial _{r_2} P^* = \dfrac{\partial _{r_2} N^*_2}{N^*_1} - \dfrac{\partial _{r_2} N^*_1}{N^*_1}P^* \end{aligned}$$
(12)

First, we study the variations of \(N^*_1\) and \(N^*_2\) according to \(r_2.\) From the definition of \(\mathbf {N}^*\), we have:

$$\begin{aligned} \left\{ \begin{array}{l} 0 = f_1(N^*_1)N^*_1 - \varepsilon _{12} N^*_1 + \varepsilon _{21} N^*_2 \\ 0 = f_2(N^*_2)N^*_2 - \varepsilon _{21} N^*_2 + \varepsilon _{12} N^*_1 \end{array}\right. . \end{aligned}$$
(13)

Let us denote \(g_1(u)=f_1(u)u\) and \(g_2(u)=f_2(u)u\). We deduce from equation (13) that \(g_1(N_1^*)=-g_2(N_2^*).\) Thus, differentiating this expression with respect to \(r_2\), we have \( \partial _{r_2} N^*_1 \partial _ug_1(N^*_1)=-\partial _{r_2} N^*_2 \partial _ug_2(N^*_2)\). Moreover, differentiating the system (13) with respect to \({r_2}\), we obtain:

$$\begin{aligned} \left\{ \begin{array}{l} 0 = \partial _{r_2} N^*_1 \partial _ug_1(N^*_1) - \varepsilon _{12}\partial _{r_2} N^*_1 + \varepsilon _{21} \partial _{r_2} N^*_2 \\ 0 = \partial _{r_2} N^*_2 \partial _ug_2(N^*_2) - \varepsilon _{21}\partial _{r_2} N^*_2 + \varepsilon _{12} \partial _{r_2} N^*_1 + N^*_2\left( 1-\dfrac{N^*_2}{K_2}\right) \end{array}\right. \end{aligned}$$
(14)

Dividing by \(N^*_1\) , we obtain:

$$\begin{aligned} \left\{ \begin{array}{l} 0 = \dfrac{\partial _{r_2} N^*_1}{N^*_1}( \partial _ug_1(N^*_1) - \varepsilon _{12}) + \varepsilon _{21} \dfrac{\partial _\varepsilon N^*_2}{N_1^*} \\ 0 =\dfrac{\partial _{r_2} N^*_2}{N^*_1}\left( \partial _ug_2(N^*_2) - \varepsilon _{21}\right) +\varepsilon _{12}\dfrac{\partial _{r_2} N^*_1}{N^*_1} + \dfrac{N^*_2}{N^*_1}\left( 1- \dfrac{N^*_2}{K_2}\right) \end{array}\right. \end{aligned}$$

From system (13), we obtain

$$\begin{aligned}f_2(N^*_2)N^*_2 = \varepsilon _{21} N^*_2 + \varepsilon _{12} N^*_1 \end{aligned}$$

and so we have: \(r_2 N^*_2\left( 1-\dfrac{N^*_2}{K_2}\right) = \varepsilon _{21} N^*_2 - \varepsilon _{12} N^*_1 \) and using \(P^* = \dfrac{N^*_1}{N^*_2}\) ,we obtain:

$$\begin{aligned} P^*\left( 1-\dfrac{N^*_2}{K_2}\right) = \dfrac{\varepsilon _{21}}{r_2}P^* -\dfrac{\varepsilon _{12}}{r_2}. \end{aligned}$$

Then we have:

$$\begin{aligned} \left\{ \begin{array}{l} 0 = \dfrac{\partial _{r_2} N^*_1}{N^*_1}( \partial _ug_1(N^*_1) - \varepsilon _{12}) + \varepsilon _{21} \dfrac{\partial _\varepsilon N^*_2}{N_1^*} \\ 0 =\dfrac{\partial _{r_2} N^*_2}{N^*_1}\left( \partial _ug_2(N^*_2) - \varepsilon _{21}\right) +\varepsilon _{12}\dfrac{\partial _{r_2} N^*_1}{N^*_1} +\dfrac{\varepsilon _{21}}{r_2}P^* -\dfrac{\varepsilon _{12}}{r_2} \end{array}\right. . \end{aligned}$$

We solve this linear system to get:

$$\begin{aligned} \left\{ \begin{array}{l} \dfrac{\partial _{r_2} N^*_1}{N^*_1} = \dfrac{ - \dfrac{\varepsilon _{21}^2}{r_2}\left( \dfrac{\varepsilon _{12}}{\varepsilon _{21}}-P^* \right) }{ ( \partial _ug_1(N^*_1) - \varepsilon _{12})(\partial _ug_2(N^*_2) - \varepsilon _{21}) - \varepsilon _{12}\varepsilon _{21}}\\ \dfrac{\partial _\varepsilon N^*_2}{N^*_1} = \dfrac{ \dfrac{\varepsilon _{21}}{r_2} \left( \dfrac{\varepsilon _{12}}{\varepsilon _{21}}-P^* \right) \left( \partial _ug_1(N^*_1)-\varepsilon _{12}\right) }{ ( \partial _ug_1(N^*_1) - \varepsilon _{12})(\partial _ug_2(N^*_2) - \varepsilon _{21}) - \varepsilon _{12}\varepsilon _{21} } \end{array}\right. . \end{aligned}$$

We deduce from equation (12) that

$$\begin{aligned} \partial _{r_2} P^* = \dfrac{ \dfrac{\varepsilon _{21}}{r_2}\left( \dfrac{\varepsilon _{12}}{\varepsilon _{21}}-P^* \right) \left( \partial _ug_1(N^*_1)-\varepsilon _{12} + \varepsilon _{21}P^* \right) }{ ( \partial _ug_1(N^*_1) - \varepsilon _{12})(\partial _ug_2(N^*_2) - \varepsilon _{21}) - \varepsilon _{12}\varepsilon _{21} } . \end{aligned}$$
(15)

To decipher the sign of \(\partial _{r_2} P^*\), we first look at the sign of the denominator \( ( \partial _ug_1(N^*_1) - \varepsilon _{12})(\partial _ug_2(N^*_2) - \varepsilon _{21}) - \varepsilon _{12}\varepsilon _{21}\)

From hypotheses (H1) and (H3), we know that \(\partial _ug_i(N^*_i)\le g_i(N^*_i)/N_i^*\). Moreover, we know from (13) that

$$\begin{aligned} \left\{ \begin{array}{l} 0=\dfrac{g_1(N_1^*)}{N_1^*} -\varepsilon _{12} +\varepsilon _{21}P^* \\ 0=\dfrac{g_2(N_2^*)}{N_2^*} -\varepsilon _{21} + \varepsilon _{12}\dfrac{1}{P^*} \end{array}\right. . \end{aligned}$$
(16)

Combining the two equations, we obtain that

$$\begin{aligned} \partial _ug_1(N^*_1) - \varepsilon _{12}\le -\varepsilon _{21} P^*<0 \ \hbox { and } \ \partial _ug_2(N^*_2) - \varepsilon _{21}\le -\varepsilon _{12} \dfrac{1}{P^*}<0 . \end{aligned}$$
(17)

Finally, we have:

$$\begin{aligned} ( \partial _ug_1(N^*_1) - \varepsilon _{12})(\partial _ug_2(N^*_2) - \varepsilon _{21}) - \varepsilon _{12}\varepsilon _{21}\ge 0. \end{aligned}$$
(18)

More precisely, we have \( ( \partial _ug_1(N^*_1) - \varepsilon _{12})(\partial _ug_2(N^*_2) - \varepsilon _{21}) - \varepsilon _{12}\varepsilon _{21}>0\) because the equality case corresponds to either \(N^*_1=0\) or \(N^*_2=0\) and we know that \(N_1^*>0\) and \(N^*_2>0.\)

Let us now look at the sign of the numerator of \(\partial _{r_2} P^*\).

We first consider the following part \(\left( \partial _ug_1(N^*_1)-\varepsilon _{12} + \varepsilon _{21}P^* \right) \). From hypotheses (H1) and (H3) combined with equation (16), we can say that \(\partial _ug_i(N^*_i)-\varepsilon _{12}\le g_i(N^*_i)/N_i^* -\varepsilon _{12}\le -\varepsilon _{21} P^* \) and then

$$\begin{aligned} \left( \partial _ug_1(N^*_1)-\varepsilon _{12} + \varepsilon _{21}P^* \right) \le 0. \end{aligned}$$
(19)

It remains to look at the sign of \(\left( \dfrac{\varepsilon _{12}}{\varepsilon _{21}}-P^*\right) \). We prove the following lemma.

Lemma 8

(Property of \(P^*\) as a function of \(r_2\)) Let \(\mathbf {N}^*\) be the solution of (13) under hypotheses (H1) and (H3). Then, the ratio \(P^*=N_2^*/N^*_1\) satisfies the following properties

  • If \(\varepsilon _{21}K_2 \ge \varepsilon _{12}K_1 \), then \( \varepsilon _{12}/\varepsilon _{21}\le P^*\) and \(P^*\) is increasing with respect to \(r_2 \).

  • If \(\varepsilon _{21}K_2 \le \varepsilon _{12}K_1 \), then \( \varepsilon _{12}/\varepsilon _{21}\ge P^*\) and \(P^*\) is decreasing with respect to \(r_2 \).

Before stating of the proof of Lemma 8, we conclude the proof of Proposition 6.

The estimates of Lemma 8 combined with the previous inequalities (15)–(19) show that the sign of \(\partial _{r_2} p^*(r_2)\) only depends on the sign of \(\left( \varepsilon _{21}K_2 - \varepsilon _{12}K_1\right) \left( p_2(0)-p_1(0)\right) \) which concludes the proof of Proposition 6.

Proof of Lemma 8

We can first show from (15)–(18)–(19) that the monotonicity of \(P^*\) depends only on the sign of \(\left( \dfrac{\varepsilon _{12}}{\varepsilon _{21}}-P^* \right) \). Moreover, from equation (17), we know that the sign of \(\left( \dfrac{\varepsilon _{12}}{\varepsilon _{21}}-P^* \right) \) is equal of the sign of \(\dfrac{g_1(N_1^*)}{N_1^*}\) and therefore depends of the position of \(N^*_1\) relative to \(K_1\).

We first start with the following changes of variables \(Y^*_1=\varepsilon _{12}N^*_1\) and \(Y^*_2=\varepsilon _{21}N^*_2\). Moreover, if we denote \(\tilde{K}_1= \varepsilon _{12}K_1\) and \(\tilde{K}_2= \varepsilon _{21}K_2\), then the system (16) becomes:

$$\begin{aligned} \left\{ \begin{array}{l} 0=\dfrac{\tilde{g}_1(Y_1^*)}{Y_1^*} -\varepsilon _{12}(1- \tilde{P}^*) \\ 0=\dfrac{\tilde{g}_2(Y_2^*)}{Y_2^*} -\varepsilon _{21}\left( 1-\dfrac{1}{\tilde{P}^*} \right) \end{array}\right. . \end{aligned}$$
(20)

where \( \tilde{g}_1(u)= u \tilde{f}_1(u)\) with \( \tilde{f}_1(u)= r_1\left( 1-\dfrac{u}{\tilde{K}_1}\right) \),   \( \tilde{g}_2(u)= u \tilde{f}_2(u)\) with \( \tilde{f}_2(u)= r_2\left( 1-\dfrac{u}{\tilde{K}_2}\right) \) and \( \tilde{P}^*=\dfrac{Y_2^*}{Y_1^*} \) .

Our aim is to show that if \(\tilde{K}_2 > \tilde{K}_1\), then \( \tilde{P}^* >1\). This is equivalent to show that \(Y_1^*> \tilde{K}_1\) if \(\tilde{K}_2 > \tilde{K}_1\). The equilibrium point \( Y^*=(Y_1^* ,Y_2^*)\) of the model (20) satisfies \(\tilde{G}_1(Y_1^*)= Y_2^*\) and \(\tilde{G}_2(Y_2^*)= Y_1^*\) where

$$\begin{aligned} \begin{array}{c} \tilde{G}_1(u)= -\dfrac{1}{\varepsilon _{12}}\left( \tilde{g}_1(u)- \varepsilon _{12}u\right) = -\dfrac{u}{\varepsilon _{12}}(\tilde{f}_1(u)-\varepsilon _{12})\\ \\ \tilde{G}_2(u)= -\dfrac{1}{\varepsilon _{12}}\left( \tilde{g}_2(u)- \varepsilon _{21}u\right) = -\dfrac{u}{\varepsilon _{21}}(\tilde{f}_2(u)-\varepsilon _{21}) . \end{array} \end{aligned}$$

First let us notice that from the hypothesis (H1), we know that \(\tilde{G}_1(\tilde{K}_1)=\tilde{K}_1\) and \(\tilde{G}_2(\tilde{K}_2)=\tilde{K}_2\). Moreover since \(\tilde{K}_2 > \tilde{K}_1\) , we can deduce from hypothesis  (H3) that

$$\begin{aligned} \begin{array}{c} \tilde{G}_1(\tilde{K}_2)= -\dfrac{\tilde{K}_2}{\varepsilon _{12}}(\tilde{f}_1(\tilde{K}_2)-\varepsilon _{12})= \tilde{K}_2 -\dfrac{\tilde{K}_2}{\varepsilon _{12}}\tilde{f}_1(\tilde{K}_2) > \tilde{K}_2 ,\\ \\ \tilde{G}_2(\tilde{K}_1)= -\dfrac{\tilde{K}_1}{\varepsilon _{21}}(\tilde{f}_2(\tilde{K}_1)-\varepsilon _{21})= \tilde{K}_1 -\dfrac{\tilde{K}_1}{\varepsilon _{21}}\tilde{f}_2(\tilde{K}_1) < \tilde{K}_1 . \end{array} \end{aligned}$$

Let us study the variation of \(\tilde{G}_1\) on \((\tilde{K}_1,+\infty )\). The derivative of \(\tilde{G}_1\) satisfies \( \tilde{G}_1^{'}(u)=1-\dfrac{r_1}{\varepsilon _{12}}\left( 1-\dfrac{2u}{\tilde{K}_1}\right) > 0\) if \(u>\tilde{K}_1\); thus, \(\tilde{G}_1\) is increasing on \((\tilde{K}_1,+\infty )\).

Let us study the variation of \(\tilde{G}_2\) on \((\tilde{K}_1,+\infty )\). As above we have: \( \tilde{G}_2^{'}(u)=1-\dfrac{r_2}{\varepsilon _{21}}\left( 1-\dfrac{2u}{\tilde{K}_2}\right) > 0\) if \(u > \dfrac{\tilde{K}_2}{2}\left( 1-\dfrac{\varepsilon _{21}}{r_2}\right) \). So, we can say that

  • If \(\dfrac{\tilde{K}_2}{2}\left( 1-\dfrac{\varepsilon _{21}}{r_2}\right) < \tilde{K}_1\), then \(\tilde{G}_2 \) is increasing on \((\tilde{K}_1,+\infty )\).

  • If \(\dfrac{\tilde{K}_2}{2}\left( 1-\dfrac{\varepsilon _{21}}{r_2}\right) > \tilde{K}_1\), then \(\tilde{G}_2 \) is decreasing on \(\left( \tilde{K}_1 , \dfrac{\tilde{K}_2}{2}\left( 1-\dfrac{\varepsilon _{21}}{r_2}\right) \right) \) and increasing on \(\left( \dfrac{\tilde{K}_2}{2}\left( 1-\dfrac{\varepsilon _{21}}{r_2}\right) , +\infty \right) . \)

From the definition of \(Y^*_1\) and \(Y^*_2\), we know that \(Y^*_1\) is a fixed point of the function \(h=\tilde{G}_2 \circ \tilde{G}_1\). Let us prove that \(Y^*_1\) belongs to \((\tilde{K}_1,\infty )\).

To conclude we need to consider two cases depending on the position of \(\tilde{K}_1\) with respect to \({\tilde{K}_2}/{2}\left( 1-{\varepsilon _{21}}/{r_2}\right) \).

On the one hand, let us assume that \({\tilde{K}_2}/{2}\left( 1-{\varepsilon _{21}}/{r_2}\right) < \tilde{K}_1\). Then, the function h is increasing on \((\tilde{K}_1,+\infty )\). Moreover, we have \(h(\tilde{K}_1)=\tilde{G}_2(\tilde{K}_1) < \tilde{K}_1 \) and \( h(\tilde{K}_2)=\tilde{G}_2 ( \tilde{G}_1 (\tilde{K}_2))> \tilde{G}_2(\tilde{K}_2) = \tilde{K}_2)\) because \(\tilde{G}_2\) is increasing on \((\tilde{K}_1,\infty )\). So, applying the intermediate value theorem to the monotone function h on \((\tilde{K}_1,+\infty )\) , there exists an unique positive real \(Y_1^* \) such that \( h(Y_1^*)=Y_1^* \) and \( Y_1^*> \tilde{K}_1 .\)

On the other hand, if \({\tilde{K}_2}/{2}\left( 1-{\varepsilon _{21}}/{r_2}\right) \ge \tilde{K}_1\), then the function h is decreasing on \((\tilde{K}_1,\tilde{K}_2/{2}\left( 1-{\varepsilon _{21}}/{r_2}\right) )\) and increasing on \((\tilde{K}_2/{2}\left( 1-{\varepsilon _{21}}/{r_2}\right) ,\infty )\). So \(h(u)<h(\tilde{K}_1)<\tilde{K}_1\) on \((\tilde{K}_1,\tilde{K}_2/{2}\left( 1-{\varepsilon _{21}}/{r_2}\right) )\). Moreover, we still have \( h(\tilde{K}_2)>\tilde{K}_2\). Thus, there exists a unique fixed point \(Y^*_1\) on \((\tilde{K}_1,\infty )\). We can then conclude that if \(\tilde{K}_2 >\tilde{K}_1\), then \(Y_1^*> \tilde{K}_1\) and \( 1 - \tilde{P}^* = \dfrac{r_1}{\varepsilon _{12}}\left( 1-\dfrac{Y_1^*}{\tilde{K}_1}\right) < 0 \) and \(P^* \) is increasing with respect to \(r_2\).

Proof of Proposition  7

Let \(\varepsilon _{12}\), \(\varepsilon _{21} \in (0,1]\) and \(\mathbf {N}^*=(N_1^*,N^*_2)\) be the equilibrium of (1). We know from Theorem 1 that for any \(p_1(0)\), \(p_2(0)\) in [0, 1], the asymptotic proportion \(p^*\) satisfies:

$$\begin{aligned} p^*(r_2) = \dfrac{p_1(0) + p_2(0)\left( \dfrac{\varepsilon _{21}}{\varepsilon _{12}}(P^*)^{2}\right) }{ 1 +\left( \dfrac{\varepsilon _{21}}{\varepsilon _{12}}(P^*)^{2}\right) } \ \hbox { with } \ P^* = \dfrac{N^*_2}{N^*_1}. \end{aligned}$$
(21)

For the sake of simplicity, we only look at the case where \(p_1(0)=1\) and \(p_2(0)=0\). The alternative case can be deduced by symmetry of the problem. Thus, we get

$$\begin{aligned} p^*(r_2) = \dfrac{1}{ 1 +\left( \dfrac{\varepsilon _{21}}{\varepsilon _{12}}(P^*)^{2}\right) }. \end{aligned}$$

We can see that \(p^*(r_2) = \frac{1}{2} \) if and only if \(P^*= \sqrt{\varepsilon _{12}/\varepsilon _{21}}\) and \(p^*\) is decreasing with respect to \(P^*\). Thus, in order to compare \(p^*(r_2)\) with 1/2, we just need to compare \(P^*\) with \(\sqrt{\varepsilon _{12}/\varepsilon _{21}}\).

First let assume that \(\varepsilon _{21} K_2>\varepsilon _{12}K_1\). Then, from the estimates of Lemma 8 we know that \(P^*\ge \varepsilon _{12}/\varepsilon _{21}\).

On the one hand, if \(\varepsilon _{12}\ge \varepsilon _{21}\), then for any \(r_2>0\) we have

$$\begin{aligned} P^*\ge \dfrac{\varepsilon _{12}}{\varepsilon _{21}}\ge \sqrt{\dfrac{\varepsilon _{12}}{\varepsilon _{21}}} \end{aligned}$$

and \(p^*(r_2)\le 1/2\) for any \(r_2 >0\).

On the other hand, if \(\varepsilon _{12}<\varepsilon _{21}\), then we have \(P^*(0)=\varepsilon _{12}/\varepsilon _{21} < \sqrt{\varepsilon _{12}/\varepsilon _{21}}\). Since \(P^*\) is increasing with respect to \(r_2\), it may exist \(\bar{r}_2>0\) such that \(P^*( \bar{r}_2) = \sqrt{\varepsilon _{12}/\varepsilon _{21}}\) (eventually \(\bar{r}_2=\infty \) if \(P^*(r_2)<\sqrt{\varepsilon _{12}/\varepsilon _{21}}\) for any \(r_2>0\)). Then, we can conclude that:

  • \(P^*(r_2)<\sqrt{\dfrac{\varepsilon _{12}}{\varepsilon _{21}}}\) and \(p^*(r_2) > \dfrac{1}{2}\), for \(r_2 <\bar{r}_2\);

  • \(P^*(r_2)=\sqrt{\dfrac{\varepsilon _{12}}{\varepsilon _{21}}}\) and \(p^*(r_2) = \dfrac{1}{2}\) , for \(r_2 =\bar{r}_2\);

  • \(P^*(r_2)>\sqrt{\dfrac{\varepsilon _{12}}{\varepsilon _{21}}}\) and \(p^*(r_2) < \dfrac{1}{2}\), for \(r_2 >\bar{r}_2\).

Let us now determine the value \(\bar{r}_2\). If \(r_2=\bar{r}_2\), then \(P^*=\sqrt{\varepsilon _{12}/\varepsilon _{21}}\) and we get the following system:

$$\begin{aligned} \left\{ \begin{array}{l} 0 = r_1\left( 1-\dfrac{N^*_1}{K_1}\right) - \varepsilon _{12} +\sqrt{\varepsilon _{21}\varepsilon _{12}} \\ 0 = \bar{r}_2\left( 1-\dfrac{N^*_2}{K_2}\right) - \varepsilon _{21} +\sqrt{\varepsilon _{21}\varepsilon _{12}} \end{array}\right. . \end{aligned}$$
(22)

Solving this system, we obtain:

$$\begin{aligned} N^*_1 = K_{1} \left[ 1-\dfrac{\varepsilon _{12}- \sqrt{\varepsilon _{21}\varepsilon _{12}}}{r_1}\right] \hbox {\, and \,} N^*_2 = K_{2} \left[ 1-\dfrac{\varepsilon _{21}- \sqrt{\varepsilon _{21}\varepsilon _{12}}}{\bar{r}_2}\right] \end{aligned}$$
(23)

Since we are in the case where \(\varepsilon _{21}< \varepsilon _{12}\), we can deduce that \(N^*_1 > K_1 \) and \(N^*_2 < K_{2} \) and therefore \(P^*(\bar{r}_2)<\dfrac{K_2}{K_1}\). So if \(\dfrac{K_2}{K_1} <\sqrt{\dfrac{\varepsilon _{12}}{\varepsilon _{21}}}\) we should have \(\bar{r}_{2}=+\infty \). In addition, using the definition of \(\bar{r}_2\) we obtain:

$$\begin{aligned} \bar{r}_2=\left\{ \begin{array}{cc} \dfrac{\sqrt{\varepsilon _{21}}[\sqrt{\varepsilon _{21}}-\sqrt{\varepsilon _{12}}]}{1-\dfrac{\varepsilon _{12}K_1}{\varepsilon _{21}K_2}\sqrt{\dfrac{\varepsilon _{21}}{\varepsilon _{12}}}\left[ 1+ \dfrac{\sqrt{\varepsilon _{12}}[\sqrt{\varepsilon _{21}}-\sqrt{\varepsilon _{12}}]}{r_1}\right] } &{} \hbox {if } \ \dfrac{K_2}{K_1} \ge \sqrt{\dfrac{\varepsilon _{12}}{\varepsilon _{21}}} \\ \infty &{} \hbox {if } \ \dfrac{K_2}{K_1} <\sqrt{\dfrac{\varepsilon _{12}}{\varepsilon _{21}}} \end{array} \right. \end{aligned}$$

Similar arguments allow to conclude for the case \(\varepsilon _{21} K_2\le \varepsilon _{12}K_1\).

A.1.2: Carrying Capacities Crucially Determine the Habitat Contributions

We now look at the effect of the migration rates on the asymptotic proportion of the fraction. In this section, we assume that the migration is symmetric:

$$\begin{aligned} \varepsilon _{12}=\varepsilon _{21}=\varepsilon \ \hbox { with } \ \varepsilon \in [0,1]. \end{aligned}$$

Under these assumptions, the asymptotic proportion of the fraction \(\mathbf {n}\) is given by the following formula:

$$\begin{aligned} p^* = \dfrac{p_1(0) + p_2(0)(P^*)^2}{ 1 + (P^*)^2} , \end{aligned}$$

where \(p_i(0)\) corresponds to the initial proportion of the fraction in each habitat i, \(p_i(0) = n_i(0)/N_i^*\) and the quantity \(P^*\) is the ratio between the density of population in habitat 2 and the one in habitat 1, \(P^*= N^*_2/N^*_1\). From this formula, we observe that the dependency of \(p^*\) according to the migration rate \(\varepsilon \) only occurs through the ratio of equilibrium densities \(P^*\).

Using this particular formula, we deduce results concerning the effect of passive migration \(\varepsilon \) on the asymptotic contribution \(p^*\) of a fraction \(\mathbf {n}\) satisfying (6).

Proposition 9

(Monotonicity of \(p^*\) with respect to \(\varepsilon \)) For any initial condition \(\mathbf {n}(0)\), the asymptotic proportion \(p^*\) associated with the fraction \(\mathbf {n}\) solving (6), is monotonic with respect to the migration rate \(\varepsilon \). More precisely, the function \(\varepsilon \mapsto p^*\) is

  • non-decreasing if \((K_1-K_2)\big (p_1(0)-p_2(0)\big )<0\);

  • constant equal to \(p^* = \dfrac{p_1(0)+p_2(0)}{2}\) if \(K_1=K_2\);

  • non-increasing if \((K_1-K_2)\big (p_1(0)-p_2(0)\big )>0\).

Our result shows that the asymptotic proportion of a given fraction v is monotonic with respect to the migration rate \(\varepsilon .\) In addition, we have proved that this monotonicity only depends on the relative repartition of the fraction inside the metapopulation patches and the carrying capacities of the habitats. We can notice that the intrinsic growth rate \(r_i\) of the habitat does not play any role in the behavior of \(p^*\) with respect to \(\varepsilon .\) In particular, even if the habitat 2 is not as good as the habitat 1, that is \(r_2<r_1\), when its carrying capacity is higher than the one of habitat 1, that is \(K_2>K_1,\) then the migration enhances the contribution from the habitat 2 which corresponds to the case where \(0=p_1(0)<p_2(0)=1\). Thus, the migration may promote the diversity from poor habitat quality if those habitats have a large carrying capacity.

To go further, we now look at the precise contribution of each habitat. More precisely, we assume that the fraction is only present in one habitat, that is either \(n_1(0)=N_1^*\) and \(n_2(0)=0\) or \(n_1(0)=0\) and \(n_2(0)=N_2^*\). In this case, we quantify the asymptotic proportion \(p^*\) with respect to 1/2 which corresponds to an equal contribution from each habitat.

Proposition 10

(Contribution of each habitat) Let us consider a fraction that fully occupies only one habitat, either \(p_1(0)=1\) and \(p_2(0)=0\) or \(p_1(0)=0\) and \(p_2(0)=1\). Then, the asymptotic proportion satisfies:

  • \(p^*<1/2\) if \((K_1-K_2)\big (p_1(0)-p_2(0)\big )<0\);

  • \(p^* = \dfrac{1}{2}\) if \(K_1=K_2\);

  • \(p^*>1/2\) if \((K_1-K_2)\big (p_1(0)-p_2(0)\big )>0\).

Our result first shows that the contribution of each habitat does not depend on the quality of the habitat \(r_i\) but only on the carrying capacity of this habitat. A habitat that can support more individuals will have a higher contribution than the other. If \(K_2>K_1\), then the contribution from the habitat 2 is always larger than 1/2 even if \(r_2<r_1.\) In this case, the contribution from the habitat 2 is larger than the contribution from the habitat 1. In this case, the habitat 2 dominates and its genetic diversity partially replaces the one from the habitat 1.

In addition, the migration cannot reverse the domination of one habitat on the other. More precisely, if \(K_2>K_1\), the contribution from habitat 2 is always higher than the one of habitat 1: \(p^* - 1/2\) does not change sign when \(\varepsilon \) varies. However, the migration tends to balance the contribution from both habitats. Indeed, proposition (10) shows that when \(K_2>K_1\), the contribution from habitat 2 tends to decrease, while the one from habitat 1 tends to increase. But in this case the contribution from habitat 2 is always higher than 1/2, while the one from habitat 1 is smaller than 1/2.

Proof of Proposition 9

Let \(\varepsilon \) be in (0, 1], \(\mathbf {F}\) satisfying hypothesis of Proposition 10, \(p_1(0)\) and \(p_2(0)\) in [0, 1] and \(\mathbf {N}^*=(N_1^*,N^*_2)\) solves  (13). We begin to study the variation of \(p^*(\varepsilon )\) with respect to \(\varepsilon \). Using Theorem 1, we have:

$$\begin{aligned} p^*(\varepsilon ) = \dfrac{p_1(0) + p_2(0)(P^*)^2}{1 + (P^*)^2} \ \hbox { where } \ P^* = \dfrac{N^*_1}{N^*_2} \end{aligned}$$

Differentiating the expression with respect to \(\varepsilon \), we obtain:

$$\begin{aligned} \partial _\varepsilon p^*(\varepsilon ) = 2\dfrac{(p_2(0)-p_1(0))}{(1 + (P^*)^2)^2}P^*\partial _\varepsilon P^*. \end{aligned}$$

The variations of \(p^*(\varepsilon )\) only depend on the sign of \((p_2(0)-p_1(0))\partial _\varepsilon P^*\). From the definition of \(P^*\), we have

$$\begin{aligned} \partial _\varepsilon P^* = \dfrac{\partial _\varepsilon N^*_2}{N^*_1} - \dfrac{\partial _\varepsilon N^*_1}{N^*_1}P^*. \end{aligned}$$
(24)

Let us now study the variations of \(N^*_1\) and \(N^*_2\) according to \(\varepsilon .\) From the definition of \(\mathbf {N}^*\), we have:

$$\begin{aligned} \left\{ \begin{array}{l} 0 = f_1(N^*_1)N^*_1 - \varepsilon N^*_1 + \varepsilon N^*_2 \\ 0 = f_2(N^*_2)N^*_2 - \varepsilon N^*_2 + \varepsilon N^*_1 \end{array}\right. \end{aligned}$$
(25)

Let us denote \(g_1(u)=f_1(u)u\) and \(g_2(u)=f_2(u)u\). We deduce from equation (13) that \(g_1(N_1^*)=\varepsilon (N^*_1-N^*_2)=-g_2(N_2^*).\) Thus, differentiating this expression with respect to \(\varepsilon \), we have \( \partial _\varepsilon N^*_1 \partial _ug_1(N^*_1)=-\partial _\varepsilon N^*_2 \partial _ug_2(N^*_2)\). Moreover, differentiating the system (13) with respect to \(\varepsilon \), we obtain

$$\begin{aligned} \left\{ \begin{array}{l} 0 = \partial _\varepsilon N^*_1 \partial _ug_1(N^*_1) - \varepsilon \partial _\varepsilon N^*_1 + \varepsilon \partial _\varepsilon N^*_2 + (N^*_2-N^*_1) \\ 0 = \partial _\varepsilon N^*_2 \partial _ug_2(N^*_2) - \varepsilon \partial _\varepsilon N^*_2 + \varepsilon \partial _\varepsilon N^*_1 + (N^*_1-N^*_2) \end{array}\right. \end{aligned}$$
(26)

Dividing by \(N^*_1\) and using \(P^* =N^*_2/N^*_1\), we obtain

$$\begin{aligned} \left\{ \begin{array}{l} 0 = \dfrac{\partial _\varepsilon N^*_1}{N^*_1}( \partial _ug_1(N^*_1) - \varepsilon ) + \varepsilon \dfrac{\partial _\varepsilon N^*_2}{N_1^*} + (P^*-1) \\ 0 =\varepsilon \dfrac{\partial _\varepsilon N^*_1}{N_1^*} + \dfrac{\partial _\varepsilon N^*_2}{N^*_1} (\partial _ug_2(N^*_2) - \varepsilon ) + (1-P^*) \end{array}\right. . \end{aligned}$$

We solve this linear system to get:

$$\begin{aligned} \left\{ \begin{array}{l} \dfrac{\partial _\varepsilon N^*_1}{N^*_1} = \dfrac{ (1-P^*)\partial _ug_2(N^*_2) }{ ( \partial _ug_1(N^*_1) - \varepsilon )(\partial _ug_2(N^*_2) - \varepsilon ) - \varepsilon ^2}\\ \dfrac{\partial _\varepsilon N^*_2}{N^*_1} = \dfrac{ (P^*-1)\partial _ug_1(N^*_1) }{ ( \partial _ug_1(N^*_1) - \varepsilon )(\partial _ug_2(N^*_2) - \varepsilon ) - \varepsilon ^2} \end{array}\right. . \end{aligned}$$

We deduce from equation (24) that

$$\begin{aligned} \partial _\varepsilon P^* = \dfrac{ (P^*-1)(\partial _ug_1(N^*_1)+\partial _ug_2(N^*_2)P^*) }{ ( \partial _ug_1(N^*_1) - \varepsilon )(\partial _ug_2(N^*_2) - \varepsilon ) - \varepsilon ^2}. \end{aligned}$$
(27)

To decipher the sign of \(\partial _\varepsilon P^*\), we first look at the sign of the denominator \(( \partial _ug_1(N^*_1) - \varepsilon )(\partial _ug_2(N^*_2) - \varepsilon ) - \varepsilon ^2\).

From hypotheses (H1) and (H3), we know that \(\partial _ug_i(N^*_i)\le g_i(N^*_i)/N_i^*\). Moreover, we know from (13) that

$$\begin{aligned} \left\{ \begin{array}{l} 0=\dfrac{g_1(N_1^*)}{N_1^*} +\varepsilon (P^*-1) \\ 0=\dfrac{g_2(N_2^*)}{N_2^*} +\varepsilon \left( \dfrac{1}{P^*}-1\right) \end{array}\right. . \end{aligned}$$
(28)

Combining the two equations, we obtain that

$$\begin{aligned} \partial _ug_1(N^*_1) - \varepsilon \le -\varepsilon P^*<0 \ \hbox { and } \ \partial _ug_2(N^*_2) - \varepsilon \le -\varepsilon \dfrac{1}{P^*}<0 . \end{aligned}$$
(29)

Finally, we have:

$$\begin{aligned} ( \partial _ug_1(N^*_1) - \varepsilon )(\partial _ug_2(N^*_2) - \varepsilon ) - \varepsilon ^2 \ge 0. \end{aligned}$$
(30)

More precisely, we have \(( \partial _ug_1(N^*_1) - \varepsilon )(\partial _ug_2(N^*_2) - \varepsilon ) - \varepsilon ^2>0\) because the equality case corresponds to either \(N^*_1\) or \(N^*_2=0\) and we know that \(N_1^*>0\) and \(N^*_2>0.\)

Let us now look at the sign of the numerator. We first consider the following part \((\partial _ug_1(N^*_1)+\partial _ug_2(N^*_2)P^*)\). Combining the two equations on (29), we get that:

$$\begin{aligned} (\partial _ug_1(N^*_1)+\partial _ug_2(N^*_2)P^*)\le 0. \end{aligned}$$
(31)

It remains to look at the sign of \((P^*-1)\). We prove the following lemma.

Lemma 11

(Property of \(P^*\)) Let \(\mathbf {N}^*\) be the solution of (13) with satisfying hypotheses (H1) and (H3). Then, the ratio \(P^*\) satisfies the following properties:

  • \(P^* < 1\) and \(P^*\) is increasing if \(K_1> K_2\).

  • \(P^* > 1\) and \(P^*\) is increasing if \(K_1< K_2\).

Before stating the proof of Lemma 11, we conclude the proof of Proposition 9. The estimates of Lemma 11 combine with the previous inequalities (29)–(30)–(31) show that the sign of \(\partial _\varepsilon p^*\) only depends on the sign of \((K_2-K_1)(p_1(0)-p_2(0))\) which concludes the proof of Proposition 9.

Proof of Lemma 11

We can first show from (27)–(30)–(31) that the monotonicity of \(P^*\) only depends on the sign of (\(P^*-1\)). Moreover, from equation (28), we know that the sign of \((P^*-1)\) is equal to the sign of \(-g_1(N_1^*)/N^*_1\) and therefore depends on the position of \(N_1^*\) relative to \(K_1\). Without loss of generality, we can assume that \(r_2 <r_1\).

In addition, we first consider the case \(K_2> K_1\). Our aim is to show that \(N_1^* >K_1\). The equilibrium point \(\mathbf {N}^*\) of the model satisfies \(G_1(N_1^*)=N^*_2 \) and \(G_2(N_2^*)=N^*_1\) where

$$\begin{aligned} G_1(u)= & {} -\dfrac{1}{\varepsilon }(g_1(u)-\varepsilon u) = -\dfrac{u}{\varepsilon }(f_1(u)-\varepsilon ) \\ G_2(u)= & {} -\dfrac{1}{\varepsilon }(g_2(u)-\varepsilon u) = -\dfrac{u}{\varepsilon }(f_2(u)-\varepsilon ). \end{aligned}$$

First let us notice that from the hypothesis (H1), we know that \(G_1(K_1) =K_1\) and \(G_2(K_2) = K_2\). Moreover, since \(K_2> K_1\), we can deduce from hypothesis (H3) that

$$\begin{aligned} G_1(K_2)= & {} - \dfrac{K_2}{\varepsilon }(f_1(K_2)-\varepsilon )= K_2 - \dfrac{K_2}{\varepsilon }f_1(K_2)>K_2 \\ G_2(K_1)= & {} - \dfrac{K_1}{\varepsilon }(f_2(K_1)-\varepsilon )= K_1 - \dfrac{K_1}{\varepsilon }f_2(K_1)<K_1. \end{aligned}$$

Then, since \(f_1\) and \(f_2\) are bounded above by \(r_1\), respectively, \(r_2\), we need to discuss with respect to the relative position of \(\varepsilon \) from \(r_1\) and \(r_2\). Let us first assume that \(\varepsilon<r_2<r_1\). We know from hypotheses H1 and H3 that fi are decreasing on \((0,\infty )\). With the assumption on \(\varepsilon \), we know that \(f_i(0)-\varepsilon = r_i-\varepsilon >0\) and \(f_i(K_i)-\varepsilon < 0\). So applying the intermediate value theorem to the monotone functions \(f_i\), there exist unique positive real \(K_{i,\varepsilon }\) in \((0,K_i)\) such that \(f_i(K_{i,\varepsilon })=\varepsilon \). Thus, we obtain that \(G_i(u)< 0\) on \((0,K_{i,\varepsilon })\) and \(G_i(u)>0\) on \((K_{i,\varepsilon },\infty )\). Moreover, we can deduce, from the definition of \(K_{i,\varepsilon }\) and the assumption on \(f_i\) that \(G_i\) is increasing on \((K_{i,\varepsilon },\infty )\). We now prove that \(N_1^*>K_1\). We can first observe that

$$\begin{aligned} G_1(K_2) = \dfrac{K_2}{\varepsilon }(\varepsilon -f_1(K_2))>K_2 = G_2^{-1}(K_2). \end{aligned}$$

Then, let us first assume that \(K_{2,\varepsilon } >K_1\). Since \(G_2\) is only positive on \((K_{2,\varepsilon },\infty )\) and increasing on \((K_{2,\varepsilon },\infty )\), we can deduce that \(G_2^{-1}\) is increasing over \((0,K_2)\) and

$$\begin{aligned} G_2^{-1}(K_1)> G_2^{-1}(0)=K_{2,\varepsilon }>K_1 = G_1(K_1). \end{aligned}$$

Since \(G_1\) and \(G_2^{-1}\) are increasing on \((K_1,K_2)\), we deduce from the intermediate value theorem that \(N^*_1\) solving \(G_1(N^*_1) = G_2^{-1}(N^*_1)\) belongs to \((K_1,K_2)\).

Let us now assume that \(K_{2,\varepsilon }<K_1\), then we have:

$$\begin{aligned} 0< G_2(K_1) = \dfrac{K_1}{\varepsilon }(\varepsilon -f_2(K_1))<K_1. \end{aligned}$$

Since \(G_2^{-1}\) is increasing on \((0,\infty )\), we deduce that

$$\begin{aligned} G_2^{-1}(K_1)>K_1=G_1(K_1). \end{aligned}$$

The same argument as above implies that \(N^*_1>K_1.\)

Let us now assume that \(r_2<\varepsilon <r_1\). In this case, \(G_2\) is positive and increasing on \([0,K_2]\). And we have that

$$\begin{aligned} 0<G_2(K_1) = \dfrac{K_1}{\varepsilon }(\varepsilon -f_2(K_1))<K_1. \end{aligned}$$

We deduce that

$$\begin{aligned} G_2^{-1}(K_1)>K_1=G_1(K_1). \end{aligned}$$
(32)

Moreover, as in the previous case, there exists \(K_{1,\varepsilon }<K_1\), such that \(G_1(K_{1, \varepsilon })= 0\) and \(G_1\) is negative on \((0,K_{1,\varepsilon })\), \(G_1\) is position on \((K_{1,\varepsilon },\infty )\). Moreover, we have:

$$\begin{aligned} G_1(K_2) = \dfrac{K_2}{\varepsilon }(\varepsilon -f_1(K_2))>K_2 = G_2^{-1}(K_2). \end{aligned}$$
(33)

The intermediate value theorem and the definition of \(N^*_1\) imply that \(N_1^* >K_1\).

Finally, let us assume that \(r_2<r_1<\varepsilon \). In this case \(G_1\) and \(G_2\) are positive and increasing on \((0,K_2)\). The previous estimates (32) and (33) hold true and we deduce that \(N^*_1>K_1.\)

In conclusion, if \(K_2>K_1\), then \(N^*_1>K_1\). Then, we deduce from equation (28) that \(P^*>1\). Conversely if \(K_2<K_1\), the same proof implies that \(P^*<1\) which concludes the proof of Lemma 11.

Proof of Proposition 10

Let \(\varepsilon \in (0,1]\) and \(\mathbf {N}^*=(N_1^*,N^*_2)\) equilibrium of (1). We know from Theorem 1 that for any \(p_1(0)\), \(p_2(0)\) in [0, 1], the asymptotic proportion \(p_\varepsilon ^*\) satisfies

$$\begin{aligned} p^* = \dfrac{p_1(0) + p_2(0)(P^*)^2}{ 1 + (P^*)^2} \ \hbox { with } \ P^* = \dfrac{N^*_2}{N^*_1}. \end{aligned}$$
(34)

We first study the case where \(p_1(0)=1\) and \(p_2(0)=0.\) From equation (34) we see that \(p^*\) is a decreasing function with respect to \(P^*.\) We thus deduce from Lemma 11 that \(p^*<1/2\) if \(K_1<K_2\) and \(p^*>1/2\) if \(K_1>K_2 .\)

Conversely, if \(p_1(0)=0\) and \(p_2(0)=1.\), from equation (34) we see that \(p^*\) is an increasing function with respect to \(P^*.\) We thus deduce from Lemma 11 that \(p^*>1/2\) if \(K_1<K_2\) and \(p^*<1/2\) if \(K_1>K_2\).

Finally, if \(K_1=K_2\), then \(P^*=1\) and \(p^*=(p_1(0) + p_2(0))/2\) which concludes the proof of Proposition 10.

A.2: Scenario 2: Source–Sink Metapopulation

We now turn to the source–sink model with possibly different migration rates between habitats. Let us remind from Theorem 1 and the equilibrium of the model that in this scenario, the asymptotic proportion of a neutral fraction with initial proportion \(p_i(0)\) in each patch, is given by the following formula:

$$\begin{aligned} p^* = \dfrac{p_1(0) + p_2(0)\dfrac{\varepsilon _{12}\varepsilon _{21}}{(r_2+\varepsilon _{21})^2}}{ 1 + \dfrac{\varepsilon _{12}\varepsilon _{21}}{(r_2+\varepsilon _{21})^2}}. \end{aligned}$$

A.2.1: High Migration from the Source Acts as a “Diversity Rescue”

From this formula, we can deduce the following properties

Proposition 12

(Trade-off between the migration rates \(\varepsilon _{ij}\)) For any initial condition \(\mathbf {n}(0)\), the asymptotic proportion \(p^*\) associated with the fraction \(\mathbf {n}\) solving (6), is monotonic with respect to migration rate from the source \(\varepsilon _{12}\), while there exists a trade-off according to the migration from the sink \(\varepsilon _{21}\). More precisely, for any \(\varepsilon _{21}\in (0,1)\), the function \(\varepsilon _{12}\mapsto p^*(\varepsilon _{12},\varepsilon _{21})\) is

  • increasing if \(p_1(0)<p_2(0)\);

  • decreasing if \(p_1(0)>p_2(0)\);

Moreover, for any \(\varepsilon _{12}\in (0,1)\), the function \(\varepsilon _{21}\mapsto p^*(\varepsilon _{12},\varepsilon _{21})\) satisfies the following properties:

  • it is decreasing if \((p_1(0)-p_2(0))(r_2-\varepsilon _{21})>0\);

  • it is increasing if \((p_1(0)-p_2(0))(r_2-\varepsilon _{21})<0\).

Our results show that migration from the source always acts in the same way. It always increases the contribution from the sink, while it reduces the contribution from the source. Indeed, if the fraction is mainly in the sink that is \(p_2(0)>p_1(0)\), then the migration from the source \(\varepsilon _{12}\) increases the asymptotic proportion \(p^*\). Conversely, if the fraction is mainly in the source that is \(p_1(0)>p_2(0)\), then the migration from the source \(\varepsilon _{12}\) decreases the asymptotic proportion \(p^*\). As in the above scenario, the migration from the good habitat tends to balance the contribution from the two habitats. However, the effect of the migration from the sink \(\varepsilon _{21}\) is nonlinear and a trade-off occurs. We see that the trade-off only depends on the quality of the sink and it does not depend on the migration from the source \(\varepsilon _{12}\) neither on the quality of the source characterized by the per capita growth rate \(r_1\) and the carrying capacity \(K_1\) of the source. For instance, if the fraction is mainly in the sink, the migration starts to increase the asymptotic proportion of the fraction from the sink. But when the migration rate reaches the threshold \(r_2\), then its effect reverses and it starts to decrease this asymptotic proportion. Thus, small migration from the sink helps fraction from the sink to contribute to the metapopulation, while a large migration from the sink reduces this contribution.

As already done in the previous scenario, we now look at the precise contribution of each habitat. More precisely, we assume that the fraction is only present in one habitat, that is either \(n_1(0)=N_1^*\) and \(n_2(0)=0\) or \(n_1(0)=0\) and \(n_2(0)=N_2^*\). In this case, we quantify the asymptotic proportion \(p^*\) by comparing it to 1/2 which corresponds to an equal contribution from each habitat.

Proposition 13

(Low migration from the source preserved source diversity) Let us consider a fraction that fully occupied only one habitat, either \(p_1(0)=1\) and \(p_2(0)=0\) or \(p_1(0)=0\) and \(p_2(0)=1\). Then, the asymptotic proportion satisfies:

  • \(p^*<1/2\) if \(\Big ( \varepsilon _{12} - \dfrac{(r_2+\varepsilon _{21})^2}{\varepsilon _{21}} \Big )\big (p_1(0)-p_2(0)\big )<0\);

  • \(p^* = \dfrac{1}{2}\) if \(\varepsilon _{12} = \dfrac{(r_2+\varepsilon _{21})^2}{\varepsilon _{21}}\);

  • \(p^*>1/2\) if \(\Big ( \varepsilon _{12} - \dfrac{(r_2+\varepsilon _{21})^2}{\varepsilon _{21}} \Big )\big (p_1(0)-p_2(0)\big )>0\).

As we could expect, the contribution of each habitat crucially depends on the migration rates \(\varepsilon _{12}\) and \(\varepsilon _{21}\), and the quality of the death rate in the source \(r_2.\) Conversely, the quality of the source does not play any role in the contribution.

We can first notice that the migration from the source plays a crucial role in the preservation of the fraction from the source. Indeed, let us compare the contribution from the sink \(p^*_{sink}\), corresponding to initial condition with \(p_1(0)=0\) and \(p_2(0)=1\) and the contribution from the source \(p^*_{source} = 1-p^*_{sink}\). From results of Proposition 13, if the migration rate from the source \(\varepsilon _{12}\) is above a threshold \(\varepsilon _c\) equal to \(\varepsilon _c=4r_2\), then, the contribution from the sink is always smaller than the contribution from the source because \(p^*_{sink}<1/2\) (see Fig. 13).

Fig. 13
figure13

(Color figure online) Behavior of the asymptotic proportion \(p^*\) as the function of the dispersal parameters \(\varepsilon _{12}\) and \(\varepsilon _{21}\): a behavior of \(p^*\) with respect to migration from the sink \(\varepsilon _{21}\) with various values of the migration rate from the source \(\varepsilon _{12}\) (dashed blue curve \(\varepsilon _{12}=0.1\), dotted-dashed red curve \(\varepsilon _{12}=0.3\), plain yellow curve \(\varepsilon _{12}=0.6\) and dotted purple curve \(\varepsilon _{12}=0.9\); b behavior of \(p^*\) with respect to both parameters \(\varepsilon _{12}\) and \(\varepsilon _{12}\). The plain black line describes the level set \(p^*=1/2\), while the red curve corresponds to the extinction limit. The black dashed line corresponds to the parameters value where the metapopulation size \(N_1^*+N_2^*\) is maximal

On top of that, our result shows that if the migration from the source is high compared to the migration from the sink, that is \(\varepsilon _{12}> (r_2+\varepsilon _{21})^2/\varepsilon _{21}\), then, the contribution from the sink is higher than the contribution from the source (\(p^*_{sink}>1/2\)). So we need large migration from the source to preserve diversity from the sink.

Proof of Proposition 12

Let \(p_1(0)\), \(p_2(0)\) be in [0, 1]. We know from Theorem 1 that if the metapopulation is composed of a source and a sink, that is \(f_i\) satisfies hypotheses (H1) and (H2), the asymptotic proportion \(p^*\) associated with \(p_1(0)\) and \(p_2(0)\) is given for any \((\varepsilon _{12},\varepsilon _{21})\in (0,1]^2\) by:

$$\begin{aligned} p^*(\varepsilon _{12},\varepsilon _{21}) = \dfrac{p_1(0) + p_2(0)\dfrac{\varepsilon _{12}\varepsilon _{21}}{(r_2+\varepsilon _{21})^2}}{ 1 + \dfrac{\varepsilon _{12}\varepsilon _{21}}{(r_2+\varepsilon _{21})^2}}. \end{aligned}$$
(35)

We first look at the behavior of \(p^*\) with respect to \(\varepsilon _{12}\). Differentiating the expression (35) with respect to \(\varepsilon _{12},\) we obtain:

$$\begin{aligned} \partial _{\varepsilon _{12}} p^*(\varepsilon _{12},\varepsilon _{21}) = \dfrac{(p_2(0) - p_1(0))\dfrac{\varepsilon _{21}}{(r_2+\varepsilon _{21})^2}}{\left( 1 + \dfrac{\varepsilon _{12}\varepsilon _{21}}{(r_2+\varepsilon _{21})^2}\right) ^2}. \end{aligned}$$
(36)

Thus, if \(p_1(0))>p_2(0)\), then \(\partial _{\varepsilon _{12}} p^*(\varepsilon _{12},\varepsilon _{21})<0\) and \(p^*\) is decreasing with respect to \(\varepsilon _{12}.\)

Let us now look at the behavior of \(p^*\) with respect to \(\varepsilon _{21}.\) Differentiating the expression (35) with respect to \(\varepsilon _{21},\) we obtain

$$\begin{aligned} \partial _{\varepsilon _{21}} p^*(\varepsilon _{12},\varepsilon _{21}) = \dfrac{(p_2(0) - p_1(0)) \dfrac{\varepsilon _{12}(r_2+\varepsilon _{21})(r_2-\varepsilon _{21})}{(r_2+\varepsilon _{21})^2} }{\left( 1 + \dfrac{\varepsilon _{12}\varepsilon _{21}}{(r_2+\varepsilon _{21})^2}\right) ^2}. \end{aligned}$$
(37)

If \(p_1(0)>p_2(0)\), then we have \(\partial _{\varepsilon _{21}} p^*(\varepsilon _{12},\varepsilon _{21})< 0\) if \(\varepsilon _{21}>r_2\) while \(\partial _{\varepsilon _{21}} p^*(\varepsilon _{12},\varepsilon _{21})> 0\) if \(\varepsilon _{21}<r_2\). This concludes the proof of Proposition 12.

Proof of Proposition 13

Let \(p_1(0)=1\) and \(p_2(0)=0\), then we know from Theorem 1 that if the metapopulation is composed of a source and a sink, that is \(f_i\) satisfies hypotheses (H1) and (H2), the asymptotic proportion \(p^*\) associated with \(p_1(0)\) and \(p_2(0)\) is given for any \((\varepsilon _{12},\varepsilon _{21})\in (0,1]^2\) by:

$$\begin{aligned} p^*(\varepsilon _{12},\varepsilon _{21}) = \dfrac{1}{ 1 + \dfrac{\varepsilon _{12}\varepsilon _{21}}{(r_2+\varepsilon _{21})^2}}. \end{aligned}$$

A direct computation shows that \(p^*\le 1/2\) is equivalent to \(\varepsilon _{12}\le (r_2+\varepsilon _{21})^2/\varepsilon _{21}.\) Conversely, \(p^*>/2\) if and only if we have \(\varepsilon _{12}> (r_2+\varepsilon _{21})^2/\varepsilon _{21}\) which concludes the proof of Proposition 13.

Appendix B: The Individual-Based Model of Neutral Genetic Fractions

We aim to compare our deterministic model with the following stochastic individual-based model. We assume that our metapopulation is composed of several individuals located in two types of habitat. Thus, we consider the spatial domain \(\left\{ 0,1\right\} \) where 0 corresponds to the first habitat and 1 to the second habitat. Each individual i is described over time t through its location \(X_{i}(t) \in \left\{ 0,1\right\} \). On top of that, we assume that our metapopulation is composed of J neutral fractions. For each subgroup \(j\in \{1,\dots ,J\}\), we denote by \(\mathbf {X}^{j}(t) = \left\{ X_{1}^{j}(t),\cdots ,X_{N_t^{j}}^{j}(t) \right\} \) the locations at time t of individuals belonging to the subgroup j, where \(N_{t}^j\) is the number of individuals at time t in the subgroup j.

Then, the location of all the individuals of the metapopulation \(\mathbf {X}\) at time t is \(\mathbf {X}(t) = \left\{ \mathbf {X}^{1}(t), \dots ,\mathbf {X}^{J}(t)\right\} =\left\{ X_{1}^{1}(t),\dots ,X_{N_{t}^1}^{1}(t),\dots ,X_{1}^{J}(t),\dots ,X_{N_{t}^J}^{J}(t) \right\} \). We use the set of all finite point measures \(\mathcal {M}\) defined by

$$\begin{aligned} \mathcal {M}= \displaystyle \left\{ \sum _{i=1}^{N}\delta _{x_{i}}\; , N \ge 0 \; , x_1 , x_2 , \cdots , x_N \in \left\{ 0,1\right\} \right\} \end{aligned}$$

. to describe the empirical density of the metapopulation given by the stochastic process:

$$\begin{aligned} \nu _t^{n} = \displaystyle \frac{1}{n}\sum _{j=1}^J\sum _{i=1}^{N_t^j}\delta _{X_i^{j}(t)} \end{aligned}$$

where \(N_t^j\) is the number of individuals of subgroup j alive at time t and \( X_i^{j}(t)\) are their location at time t. The parameter n is the typical size of the population which eventually tends toward \(\infty \). The dynamics of the metapopulation is described by the following process:

  1. 1.

    The initial distribution \(\nu _0^n \in \mathcal {M}\) is given by \( \displaystyle \nu _0^n=\frac{1}{n} \sum _{j=1}^J\sum \nolimits _{i=1}^{N_0^j} \delta _{X_i^{j}(0)} \) where \(\displaystyle \sum \nolimits _{j=1}^J N_0^j(0)=n.\)

  2. 2.

    For each individual located at \(x \in \left\{ 0,1\right\} \) , we define three independent exponential clocks as follows:

    • reproduction rate b(x)

    • natural death rate d(x)

    • competition mortality rate \(\displaystyle \dfrac{b(x)-d(x)}{n K(x)}\sum _{j=1}^J\sum \nolimits _{i=1}^{N_t^j}\delta _{x}(X_j^i(t))\) where K(x) is the carrying capacity in location x and \(U(x,y)=0\) if \(x\ne y\) and \(U(x,x)=1\).

  3. 3.

    If an individual dies, it disappears definitively.

  4. 4.

    Each individual generates descendants that either remain in the parent location x or move to an other location y with a probability \(\varepsilon (x,y)\) which depends on the parent location x and its new location y.

In our model, we see that the process \(\nu _t^{n}\) is the sum of J processes \(\nu _t^{j,n}\) corresponding to each subgroup and defined by \(\nu _t^{j,n} = \displaystyle \frac{1}{n}\sum \nolimits _{i=1}^{N_{j}(t)}\delta _{X_{i}^{j}(t)} \), starting from \(\nu _0^{j,n} = \displaystyle \frac{1}{n}\sum \nolimits _{i=1}^{N_{j}(0)}\delta _{X_{i}^{j}(0)} \).

We can deduce the infinitesimal generator \(L^n\) for the entire population process \(\nu ^n_t\) defined for a large class of functions \(\phi \) from \(\mathcal {M}\) into \(\mathbb {R}\) by:

$$\begin{aligned} L^{n} \phi (\nu )= & {} \displaystyle n\sum _{x=0}^{1}\nu (x)\sum _{z=0}^{1}\varepsilon (x,z)\left[ \phi \left( \nu +\frac{1}{n}\delta _{x+z}\right) -\phi (\nu )\right] b(x) \\&\displaystyle +\, n\sum _{x=0}^{1}\nu (x)\left[ \phi \left( \nu -\frac{1}{n}\delta _x\right) -\phi (\nu )\right] \left\{ d(x)+ \frac{b(x)-d(x)}{K(x)}\nu (x)\right\} . \end{aligned}$$

Then using classical results on stochastic process (Bansaye and Méléard 2015),

we can deduce that the process \((\nu ^{n}_t)\) converges in law to the deterministic continuous functions \(\mathbf {N}(t)=(N_1(t),N_2(t))\) solution of the following ODE systems:

$$\begin{aligned} \left\{ \begin{array}{c} N_1^{'} =r_1 N_1 \left( 1-\dfrac{N_1}{K_1}\right) - b_1 \varepsilon _{12}N_1 + b_2 \varepsilon _{21}N_2\\ N_2^{'} =r_2 N_2 \left( 1-\dfrac{N_2}{K_2}\right) - b_2 \varepsilon _{21}N_2 + b_1 \varepsilon _{12}N_1 \end{array} \right. . \end{aligned}$$

Knowing the process \(\nu ^n_t\), we can define the infinitesimal generator \(L_t^{j,n}\) for each subgroup process \(\nu ^{j,n}_t\) as follows:

$$\begin{aligned} L_t^{j,n} \phi (\mu )= & {} \displaystyle n\sum _{x=0}^{1}\mu (x)\sum _{z=0}^{1}\varepsilon (x,z)\left[ \phi \left( \mu +\frac{1}{n}\delta _{x+z}\right) -\phi (\mu )\right] b(x) \\&\displaystyle + n\sum _{x=0}^{1}\mu (x)\left[ \phi \left( \mu -\frac{1}{n}\delta _x\right) -\phi (\mu )\right] \left\{ d(x)+ \frac{b(x)-d(x)}{K(x)}^1\nu ^n_t(x)\right\} . \end{aligned}$$

We deduce that each process \((\nu ^{j,n}_t)\) whose infinitesimal generator depends on \(\nu ^n_t\), converges in law to the deterministic continuous functions \(\mathbf {n}_{t}^{j} \) where \(\mathbf {n}_{t}^{j} =(n_{1}^{j}(t),n_{2}^{j}(t))\) solves the ODEs system

$$\begin{aligned} \left\{ \begin{array}{c} n_1' =r_1 n_1 \left( 1-\dfrac{N_1}{K_1}\right) - b_1 \varepsilon _{12}n_1 + b_2 \varepsilon _{21}n_2\\ n_2' =r_2 n_2 \left( 1-\dfrac{N_2}{K_2}\right) - b_2 \varepsilon _{21}n_2 + b_1 \varepsilon _{12}n_1 \end{array} \right. . \end{aligned}$$
(38)

Numerical simulations. We compare our analytical result associated with the ODE model (38) with two of the individual-based model: the asymptotic proportion \(p^*\) of fractions in each habitat and the asymptotic diversity index Div (see Fig. 14). We assume that the metapopulation is composed of \(J=2\) fractions initially located, respectively, in habitat 1 and 2. We observe that our analytical results fit well with both the temporal dynamics of the proportions and the dependence to parameters to asymptotic proportion and diversity even if the typical number of individuals in the individual-based model is relatively small \(n=10\).

Fig. 14
figure14

(Color figure online) Stochastic and deterministic asymptotic proportion \(p^*\) (a) and diversity Div (b) of a metapopulation composed of 2 fractions. The dotted lines corresponds to our analytical formulas (10) (a) and (8) (b). The circles correspond to the median of, respectively, the asymptotic proportion (a) and the asymptotic diversity (b) of the IBM model averaged over \(10^3\) replicates (\(n=10\) individuals). Shading envelope is interval between 0.01 and 0.99 quantiles of, respectively, the asymptotic proportion and diversity obtained from the individual-based model

Fig. 15
figure15

(Color figure online) Evolution of the two fractions \(\mathbf {n}_1\) and \(\mathbf {n}_2\) inside the metapopulation \(\mathbf {N}=(N_1 ,N_2)\) composed of two favorable habitats (\(r_1=0.3\) and \(r_2=0.1\)) with various carrying capacities: ac \(K_1=300\) and \(K_2=100\); bd \(K_1=300\) and \(K_2=900\)

Appendix C: Numerical Simulations for Initially Isolated Population

C.1: Symmetric Migration Always Promotes Diversity

We first assume that the migrations between habitats are identical \(\varepsilon _{12}=\varepsilon _{21}=\varepsilon \) with \(\varepsilon \) in [0, 1]. We investigate two cases: 1) where \(K_1 > K_2\) and 2) where \(K_1 < K_2\). In the first case, habitat 1 is better than habitat 2 because both the per capita growth rate and the carrying capacity of habitat 1 are larger than that of habitat 2. Conversely, in the second case, habitat 1 has a higher per capita growth rate, but habitat 2 has a larger carrying capacity.

We first observe that even if the two subpopulations are not initially at equilibrium, the fractions spread and persist in all the habitats. For instance, fraction 1 is not present initially in habitat 2 but its density will rapidly grow to a positive equilibrium (see Fig. 15a–b and d–e. So, even if the metapopulation is not at equilibrium, the richness of the genetic fraction is preserved at global scale as well as the local scale. Moreover, we can see from Fig. 15c, f that in both cases the proportions of each fraction in each habitat will converge to a proportion which is the same over the habitats. As in the equilibrium case, we observe that the asymptotic proportion of each fraction crucially depends on the ratio between the two carrying capacities.

C.2: Directional Migration (\(\varepsilon _{12} \ne \varepsilon _{21}\))

We now look at the effect of a directional migration on the dynamics of neutral genetic diversity. We may assume that migration \(\varepsilon _{12}\) from habitat 1 might be different from migration \(\varepsilon _{21}\) from habitat 2. As above, we first show that the proportion of each fraction \(\mathbf {p}_1\) and \(\mathbf {p}_2\) converges to asymptotic quantities \(p^*_1\), respectively, \(p^*_2\) which are the same in both habitats (see Fig. 16). We can first notice that the qualitative behavior of the asymptotic proportion \(p^*\) does not truly depend on the carrying capacities. Indeed, we observe from Fig. 16 that the migration from the lower-quality habitat 2 tends to decrease the contribution from the better-quality habitat 1, while migration from the better habitat 1 reinforces its contribution.

Fig. 16
figure16

(Color figure online) Behavior of the asymptotic proportion of the fraction 1 \(\mathbf {p}_1\) with respect to migration rate \(\varepsilon _{12}\) and \(\varepsilon _{21}\) for different quality habitats: a \(K_1 =1000\); \(K_2=100\) b \(K_1 =K_2 =500\) and c \(K_1=100\); \(K_2=1000\). The black line corresponds to the level set \(p_1=0.5\). The fraction is initially in habitat 1 and \(r_1=0.3\), \(r_2=0.1\)

Fig. 17
figure17

(Color figure online) Behavior of the \(\gamma \)-diversity as a function of the migrations rates \(\varepsilon _{12}\) and \(\varepsilon _{21}\) for various carrying capacities: a \(K_1 =1000\), \(K_2=100\), b \(K_1 =K_2 =500\) and c \(K_1=100\), \(K_2=1000\). Metapopulation is initially composed of 2 fractions, one in habitat 1 and the other in habitat 2. Habitat qualities are stated to \(r_1=0.3\), \(r_2=0.1\). The plain black line corresponds to the case where the proportion of the fraction 1 is equal to 0.5

However, the strength of the contribution of the habitat depends on the relative difference of carrying capacities \(K_1\) and \(K_2\) . More precisely, if \(K_1>K_2\) which means that habitat 1 is bigger than habitat 2, then if the migration from habitat 2 is not so high enough, then the contribution from habitat 1 is always bigger than the one from habitat 2. Indeed, we can see from Fig. 16 that \(p^*>1/2\) for \(\varepsilon _{21} <0.2\) and any \(\varepsilon _{12}.\) Conversely, if \(K_1< K_2\), a low migration from habitat 1 ( \(\varepsilon _{12}< 0.1\)) implies that contribution from habitat 2 is higher than the one from habitat 1. Even if the qualitative behavior seems symmetric with respect to \(K_1\) and \(K_2\), the thresholds are different.

We now look at the effect of migration on the local and global genetic diversity indices.

We can first notice that the behavior of the diversity with respect to the dispersal parameters is no more monotonic. We can observe from Fig. 17 that for any carrying capacities, the diversity first increases with respect to either migration from the habitat 1 or 2 until a threshold which corresponds to the value \(p^*=1/2\) and then it may decrease. We can observe that the high level of diversity is reached when migration rates are of the same order, conversely to the case of source–sink model where the asymmetry of habitats forces an asymmetry in the migration. However, a small asymmetry occurs if we look closely to the maximum of diversity.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Garnier, J., Lafontaine, P. Dispersal and Good Habitat Quality Promote Neutral Genetic Diversity in Metapopulations. Bull Math Biol 83, 20 (2021). https://doi.org/10.1007/s11538-020-00853-5

Download citation

Keywords

  • Neutral genetic diversity
  • Metapopulation
  • Dynamical system
  • Inside dynamics

Mathematics Subject Classification

  • 37N25
  • 34D23
  • 92D25
  • 92D40