Inside Dynamics of Integrodifference Equations with Mutations

  • 59 Accesses


The method of inside dynamics provides a theory that can track the dynamics of neutral gene fractions in spreading populations. However, the role of mutations has so far been absent in the study of the gene flow of neutral fractions via inside dynamics. Using integrodifference equations, we develop a neutral genetic mutation model by extending a previously established scalar inside dynamics model. To classify the mutation dynamics, we define a mutation class as the set of neutral fractions that can mutate into one another. We show that the spread of neutral genetic fractions is dependent on the leading edge of population as well as the structure of the mutation matrix. Specifically, we show that the neutral fractions that contribute to the spread of the population must belong to the same mutation class as the neutral fraction found in the leading edge of the population. We prove that the asymptotic proportion of individuals at the leading edge of the population spread is given by the dominant right eigenvector of the associated mutation matrix, independent of growth and dispersal parameters. In addition, we provide numerical simulations to demonstrate our mathematical results, to extend their generality and to develop new conjectures about our model.

This is a preview of subscription content, log in to check access.

Access options

Buy single article

Instant unlimited access to the full article PDF.

US$ 39.95

Price includes VAT for USA

Subscribe to journal

Immediate online access to all issues from 2019. Subscription will auto renew annually.

US$ 99

This is the net price. Taxes to be calculated in checkout.

Fig. 1
Fig. 2
Fig. 3
Fig. 4


  1. Arenas M (2015) Trends in substitution models of molecular evolution. Front Genet 6:319

  2. Bateman AW, Buttenschön A, Erickson KD, Marculis NG (2017) Barnacles vs bullies: modelling biocontrol of the invasive european green crab using a castrating barnacle parasite. Theor Ecol 10(3):305–318

  3. Bonnefon O, Garnier J, Hamel F, Roques L (2013) Inside dynamics of delayed travelling waves. Math Mod Nat Phen 8:44–61

  4. Bonnefon O, Coville J, Garnier J, Roques L (2014) Inside dynamics of solutions of integro-differential equations. Discret Contin Dyn Syst Ser B 19(10):3057–3085

  5. Bromham L, Penny D (2003) The modern molecular clock. Nat Rev Genet 4(3):216

  6. Cooley JW, Tukey JW (1965) An algorithm for the machine calculation of complex Fourier series. Math Comput 19(90):297–301

  7. Duret L (2008) Neutral theory: the null hypothesis of molecular evolution. Nat Educ 1:803–806

  8. Edmonds CA, Lillie AS, Cavalli-Sforza LL (2004) Mutations arising in the wave front of an expanding population. Proc Natl Acad Sci 101(4):975–979

  9. Excoffier L, Ray N (2008) Surfing during population expansions promotes genetic revolutions and structuration. Trends Ecol Evol 23(7):347–351

  10. Excoffier L, Foll M, Petit RJ (2009) Genetic consequences of range expansions. Annu Rev of Ecol Evol Syst 40:481–501

  11. Felsenstein J (1981) Evolutionary trees from DNA sequences: a maximum likelihood approach. J Mol Evol 17(6):368–376

  12. Garnier J, Lewis MA (2016) Expansion under climate change: the genetic consequences. Bull Math Biol 78(11):2165–2185

  13. Garnier J, Giletti T, Hamel F, Roques L (2012) Inside dynamics of pulled and pushed fronts. J Math Pures Appl 98(4):428–449

  14. Hallatschek O, Nelson DR (2008) Gene surfing in expanding populations. Theor Popul Biol 73(1):158–170

  15. Hallatschek O, Hersen P, Ramanathan S, Nelson DR (2007) Genetic drift at expanding frontiers promotes gene segregation. Proc Natl Acad Sci 104(50):19926–19930

  16. Ho S (2008) The molecular clock and estimating species divergence. Nat Educ 1(1):1–2

  17. Jukes T, Cantor C (1969) Evolution of protein molecules. In: ‘mammalian protein metabolism’. (ed. hn munro.) pp 21–132

  18. Kimura M (1980) A simple method for estimating evolutionary rates of base substitutions through comparative studies of nucleotide sequences. J Mol Evol 16(2):111–120

  19. Klopfstein S, Currat M, Excoffier L (2006) The fate of mutations surfing on the wave of a range expansion. Mol Biol Evol 23(3):482–490

  20. Kot M, Lewis MA, van den Driessche P (1996) Dispersal data and the spread of invading organisms. Ecology 77(7):2027–2042

  21. Krkošek M, Lauzon-Guay JS, Lewis MA (2007) Relating dispersal and range expansion of California sea otters. Theor Popul Biol 71(4):401–407

  22. Lande R (1992) Neutral theory of quantitative genetic variance in an island model with local extinction and colonization. Evolution 46(2):381–389

  23. Lehe R, Hallatschek O, Peliti L (2012) The rate of beneficial mutations surfing on the wave of a range expansion. PLoS Comput Biol 8(3):e1002447

  24. Lewis MA, Petrovskii SV, Potts JR (2016) The mathematics behind biological invasions, vol 44. Springer, Cham

  25. Lewis MA, Marculis NG, Shen Z (2018) Integrodifference equations in the presence of climate change: persistence criterion, travelling waves and inside dynamics. J Math Biol 77(6–7):1649–1687

  26. Li B, Weinberger HF, Lewis MA (2005) Spreading speeds as slowest wave speeds for cooperative systems. Math Biosci 196(1):82–98

  27. Lui R (1983) Existence and stability of travelling wave solutions of a nonlinear integral operator. J Math Biol 16(3):199–220

  28. Lutscher F, Pachepsky E, Lewis MA (2005) The effect of dispersal patterns on stream populations. SIAM Rev 47(4):749–772

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

  30. Marculis NG, Lui R, Lewis MA (2017) Neutral genetic patterns for expanding populations with nonoverlapping generations. Bull Math Biol 79(4):828–852

  31. Marculis NG, Garnier J, Lui R, Lewis MA (2019) Inside dynamics for stage-structured integrodifference equations. In Press, J Math Biol

  32. Mayr E (1940) Speciation phenomena in birds. Am Nat 74(752):249–278

  33. Morin PA, Luikart G, Wayne RK (2004) Snps in ecology, evolution and conservation. Trends Ecol Evol 19(4):208–216

  34. Pannell JR, Charlesworth B (1999) Neutral genetic diversity in a metapopulation with recurrent local extinction and recolonization. Evolution 53(3):664–676

  35. Pannell JR, Charlesworth B (2000) Effects of metapopulation processes on measures of genetic diversity. Philos Trans R Soc Lond B Biol Sci 355(1404):1851–1864

  36. Reimer JR, Bonsall MB, Maini PK (2016) Approximating the critical domain size of integrodifference equations. Bull Math Biol 78(1):72–109

  37. Roques L, Garnier J, Hamel F, Klein EK (2012) Allee effect promotes diversity in traveling waves of colonization. Proc Natl Acad Sci 109(23):8828–8833

  38. Roques L, Hosono Y, Bonnefon O, Boivin T (2015) The effect of competition on the neutral intraspecific diversity of invasive species. J Math Biol 71(2):465–489

  39. Selkoe KA, Toonen RJ (2006) Microsatellites for ecologists: a practical guide to using and evaluating microsatellite markers. Ecol Lett 9(5):615–629

  40. Slatkin M (1985) Gene flow in natural populations. Annu Rev Ecol Syst 16(1):393–430

  41. Slatkin M, Excoffier L (2012) Serial founder effects during range expansion: a spatial analog of genetic drift. Genetics 191(1):171–181

  42. Stokes A (1976) On two types of moving front in quasilinear diffusion. Math Biosci 31(3–4):307–315

  43. Van Kirk RW, Lewis MA (1997) Integrodifference models for persistence in fragmented habitats. Bull Math Biol 59(1):107

  44. Weinberger HF (1982) Long-time behavior of a class of biological models. SIAM J Math Anal 13(3):353–396

  45. Zhou Y, Kot M (2011) Discrete-time growth-dispersal models with shifting species ranges. Theor Ecol 4(1):13–25

Download references


This research was supported by a grant to MAL from the Natural Science and Engineering Research Council of Canada (Grant No. NET GP 434810-12) to the TRIA Network, with contributions from Alberta Agriculture and Forestry, Foothills Research Institute, Manitoba Conservation and Water Stewardship, Natural Resources Canada-Canadian Forest Service, Northwest Territories Environment and Natural Resources, Ontario Ministry of Natural Resources and Forestry, Saskatchewan Ministry of Environment, West Fraser and Weyerhaeuser. MAL is also grateful for support through NSERC and the Canada Research Chair Program. NGM acknowledges support from NSERC TRIA-Net Collaborative Research Grant and would like to express his thanks to the Lewis Research Group for the many discussions and constructive feedback throughout this work. We are also grateful for the helpful and detailed comments from anonymous reviewers.

Author information

Correspondence to Nathan G. Marculis.

Additional information

Publisher's Note

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


Derivation of a General Mutation Matrix

Here, we show how one can generalize the assumption of a single locus with n different neutral alleles to m loci with two neutral alleles. Let there be m independent loci \(a_i\), \(1 \le i \le m\), where each loci has one of the two possible alleles: \(a_i=0\) or \(a_i=1\). Then we define the transition probabilities as follows:

$$\begin{aligned} \text {Pr}\{a_i=0 \rightarrow a_i=1\} = q_i \quad \text {and}\quad \text {Pr}\{a_i=1 \rightarrow a_i=0\} = r_i. \end{aligned}$$

We index this process by \(t \in {\mathbb {N}}\) where t describes the number of possible transitions taken so far. There are \(2^m\) possible states for this system. Let \(n=2^m\) and let the probability of being in state j, \(1 \le j \le n\), be given by \(v^j\) where the state is

$$\begin{aligned} j = 1 + \sum _{i=1}^m a_i2^{i-1}. \end{aligned}$$

In the case when \(m=2\), there are four total states. We denote our states in the following form,

$$\begin{aligned} \begin{pmatrix} a_1\\ a_2 \end{pmatrix}. \end{aligned}$$

Our indexing for j gives the following relationship between the state and the index as follows:


By letting \(m_{jl} = \text {Pr}\{v^j \rightarrow v^l\}\), then the mutation matrix becomes

$$\begin{aligned} {\mathbf {M}} = \begin{bmatrix} (1-q_1)(1-q_2)&\quad r_1(1-q_2)&\quad (1-q_1)r_2&\quad r_1r_2\\ q_1(1-q_2)&\quad (1-r_1)(1-q_2)&\quad q_1r_2&\quad (1-r_1)r_2\\ (1-q_1)q_2&\quad r_1q_2&\quad (1-q_1)(1-r_2)&\quad r_1(1-r_2)\\ q_1q_2&\quad (1-r_1)q_2&\quad q_1(1-r_2)&\quad (1-r_1)(1-r_2) \end{bmatrix}. \end{aligned}$$

From this example, one can deduce how to generalize this process for more than two neutral alleles, making the structure of this mutation matrix quite general.

Proofs of the Theorems

Proof of Theorem 1

Without loss of generality, we can assume that neutral fraction i belongs to the mutation class q. Then, since \({\mathbf {M}}\) is block diagonal, we only need to consider the following equation

$$\begin{aligned} {\mathbf {v}}_{q,t+1}(x) = {\mathbf {M}}_q \int _{-\infty }^\infty k(x-y)g(u_t(y)){\mathbf {v}}_{q,t}(y)\,\text {d}y. \end{aligned}$$

Using the fact that \(0<g(u) \le g(0)\) for all \(u \in (0,1)\) we can use a comparison principle, see Lemma 2.1 of Li et al. (2005), to show that a new sequence \({\mathbf {w}}_{q,t}(x)\) defined by

$$\begin{aligned} {\mathbf {w}}_{q,t+1}(x) = g(0){\mathbf {M}}_q \int _{-\infty }^\infty k(x-y){\mathbf {w}}_{q,t}(y)\,\text {d}y \end{aligned}$$

is always greater than the solution to any neutral fraction, \({\mathbf {v}}_{q,t}(x)\), with the same initial condition \({\mathbf {w}}_{q,0}(x)= {\mathbf {v}}_{q,0}(x)\). The solution of (32) is given by the t-fold convolution

$$\begin{aligned} {\mathbf {w}}_{q,t}(x) = \left[ g(0){\mathbf {M}}_q\right] ^t k^{*t}{\mathbf {w}}_{q,0}(x). \end{aligned}$$

Applying the reflected bilateral Laplace transform to (33) and using the convolution theorem, we obtain

$$\begin{aligned} {\mathcal {M}}[{\mathbf {w}}_{q,t}(x)](s)&= [g(0){\mathbf {M}}_q]^{t}\left[ {\mathcal {M}}\left[ k(x)\right] (s)\right] ^{t}{\mathcal {M}}\left[ {\mathbf {w}}_{q,0}(x)\right] (s) \end{aligned}$$
$$\begin{aligned}&= [g(0){\mathbf {M}}_q]^{t}\left[ \text {e}^{\frac{\sigma ^2s^2}{2}+\mu s}\right] ^{t}{\mathcal {M}}\left[ {\mathbf {w}}_{q,0}(x)\right] (s) \end{aligned}$$
$$\begin{aligned}&= [g(0){\mathbf {M}}_q]^{t}\text {e}^{\frac{\sigma ^2ts^2}{2}+\mu ts}{\mathcal {M}}\left[ {\mathbf {w}}_{q,0}(x)\right] (s) \end{aligned}$$
$$\begin{aligned}&= [g(0){\mathbf {M}}_q]^{t}{\mathcal {M}}\left[ \frac{1}{\sqrt{2\pi \sigma ^2t}}\text {e}^{-\frac{(x-\mu t)^2}{2\sigma ^2t}}\right] (s){\mathcal {M}}\left[ {\mathbf {w}}_{q,0}(x)\right] (s) \end{aligned}$$
$$\begin{aligned}&= [g(0){\mathbf {M}}_q]^{t}{\mathcal {M}}\left[ (k_t*{\mathbf {w}}_{q,0})(x)\right] (s), \end{aligned}$$

where \(k_t\) is Gaussian with mean \(\mu t\) and variance \(\sigma ^2t\). Then applying the inverse transform yields

$$\begin{aligned} {\mathbf {w}}_{q,t}(x)&= [g(0){\mathbf {M}}_q]^t(k_t*{\mathbf {w}}_{q,0})(x) \end{aligned}$$
$$\begin{aligned}&= [g(0){\mathbf {M}}_q]^t\int _{-\infty }^\infty \frac{1}{\sqrt{2\pi \sigma ^2t}}\text {e}^{-\frac{(x-y-\mu t)^2}{2\sigma ^2t}}{\mathbf {w}}_{q,0}(y)\,\text {d}y. \end{aligned}$$

In the moving half-frame with fixed \(A \in {\mathbb {R}}\), consider the element \(x_0+ct\) with \(c \ge c^* = \sqrt{2\sigma ^2\ln (g(0))} + \mu \). When we rewrite \({\mathbf {w}}_{q,t}(x)\) in this moving half-frame, we have

$$\begin{aligned} {\mathbf {w}}_{q,t}(x_0+ct)&= [g(0){\mathbf {M}}_q]^t\int _{-\infty }^\infty \frac{1}{\sqrt{2\pi \sigma ^2t}}\text {e}^{-\frac{(x_0+ct-y-\mu t)^2}{2\sigma ^2t}}{\mathbf {w}}_{q,0}(y)\,\text {d}y. \end{aligned}$$

Expanding in the exponential yields

$$\begin{aligned} \frac{(x_0+ct-y-\mu t)^2}{2\sigma ^2t}&=\frac{(x_0-y)^2}{2\sigma ^2t} + \frac{2(c-\mu )t(x_0-y)+(c-\mu )^2t^2}{2\sigma ^2t} \end{aligned}$$
$$\begin{aligned}&\ge \frac{(x_0-y)^2}{2\sigma ^2t} + \frac{c-\mu }{\sigma ^2}(x_0-y) + \ln (g(0))t. \end{aligned}$$


$$\begin{aligned} {\mathbf {w}}_{q,t}(x_0+ct)&\le {\mathbf {M}}_q^t\frac{\text {e}^{\ln (g(0))t}}{\sqrt{2\pi \sigma ^2t}}\int _{-\infty }^\infty \text {e}^{-\frac{(x_0-y)^2}{2\sigma ^2t}}\text {e}^{-\frac{c-\mu }{\sigma ^2}(x_0-y)}\text {e}^{-\ln (g(0))t} {\mathbf {w}}_{q,0}(y)\,\text {d}y \end{aligned}$$
$$\begin{aligned}&={\mathbf {M}}_q^t\frac{1}{\sqrt{2\pi \sigma ^2t}}\int _{-\infty }^\infty \text {e}^{-\frac{(x_0-y)^2}{2\sigma ^2t}}\text {e}^{-\frac{c-\mu }{\sigma ^2}(x_0-y)}{\mathbf {w}}_{q,0}(y)\,\text {d}y \end{aligned}$$
$$\begin{aligned}&={\mathbf {M}}_q^t\frac{\text {e}^{-\frac{c-\mu }{\sigma ^2}x_0}}{\sqrt{2\pi \sigma ^2t}}\int _{-\infty }^\infty \text {e}^{-\frac{(x_0-y)^2}{2\sigma ^2t}}\text {e}^{\frac{c-\mu }{\sigma ^2}y}{\mathbf {w}}_{q,0}(y)\,\text {d}y. \end{aligned}$$

Since \(x_0 \ge A\), we have

$$\begin{aligned} {\mathbf {w}}_{q,t}(x_0+ct)&\le {\mathbf {M}}_q^t\frac{\text {e}^{-\frac{A(c-\mu )}{\sigma ^2}}}{\sqrt{2\pi \sigma ^2t}}\int _{-\infty }^\infty \text {e}^{\frac{c-\mu }{\sigma ^2}y}{\mathbf {w}}_{q,0}(y)\,\text {d}y. \end{aligned}$$

Since \({\mathbf {M}}_q\) is a Markov matrix, we know that \(\lim _{t\rightarrow \infty } {\mathbf {M}}_q^t = [{\mathbf {r}}_q, \ldots , {\mathbf {r}}_q]\) where \({\mathbf {r}}_q\) is the right eigenvector of \({\mathbf {M}}_q\) corresponding to eigenvalue 1 such that \(\sum _{i=1} r_{q,i} = 1\). By Assumption A4, \(\int _{-\infty }^\infty \text {e}^{\frac{c-\mu }{\sigma ^2}y}w^i_{q,0}(y)\,\text {d}y < \infty \) for every i in mutation class q we have \(w^i_{q,t}(x_0+ct) \rightarrow 0\) uniformly as \(t\rightarrow \infty \) in \([A,\infty )\). Recall that \(w_q^i\) was constructed so that \(0 \le v^i_{q,t}(x) \le w^i_{q,t}(x)\). This implies the uniform convergence of \(v^i_t(x) \rightarrow 0\) as \(t\rightarrow \infty \) in the moving half-frame \([A +ct, \infty )\) for each i in mutation class q. The proof of Theorem 1 is complete.\(\square \)

Proof of Theorem 2

Repeat the proof of Theorem 1 in the left moving half-frame with fixed \(A \in {\mathbb {R}}\) and consider the element \(x_0-ct\) with \(c \ge c^*_- = \sqrt{2\sigma ^2\ln (g(0))}-\mu \). From this change, the result follows in the same manner as in Theorem 1.\(\square \)

Proof of Theorem 3

We begin by decomposing \({\mathbb {R}}^n\) according to the eigenspace of the matrix \({\mathbf {M}}_q\). By assumption, since all the blocks in \({\mathbf {M}}\) are primitive, we know that the principle eigenvalue is simple and equal to 1 with nonnegative eigenvector \({\mathbf {r}}_q\) because \({\mathbf {M}}_q\) is Markov. With a small abuse of notation, we call \({\mathbf {r}}_q\) the eigenvector of \({\mathbf {M}}\) associated with the eigenvalue 1 coming from the matrix \({\mathbf {M}}_q\). In a similar manner, we define \(\varvec{\ell }_q\) be the eigenvector of \({\mathbf {M}}^T\) associated with the eigenvalue 1 from the matrix \({\mathbf {M}}_q^T\). Moreover, since the mutation matrix \({\mathbf {M}}\) is block diagonal, we can decompose the space \({\mathbb {R}}^n\) as follows

$$\begin{aligned} {\mathbb {R}}^n = \bigoplus _{q=1}^{n_q}{\mathbf {r}}_q{\mathbb {R}}\bigoplus _{q=1}^{n_q}\left( {\mathbf {r}}_q{\mathbb {R}}\right) ^{\perp }, \end{aligned}$$

where \(\left( {\mathbf {r}}_q{\mathbb {R}}\right) ^{\perp } = \left\{ {\mathbf {v}} \in {\mathbb {R}}^n, {\mathbf {v}}^T\varvec{\ell }_q={\mathbf {0}}\right\} \). Let \({\mathbf {v}}_0(x)\) satisfy \(0 \le v^i_0(x) \le {u}_0(x)\) in \({\mathbb {R}}\) for \(i=1,\ldots , n\). Then, we can decompose \({\mathbf {v}}_t(x)\) as follows

$$\begin{aligned} {\mathbf {v}}_t(x) = \sum _{q=1}^{n_q}a_t^q(x){\mathbf {r}}_q + \sum _{q=1}^{n_q}b_t^q(x){\mathbf {h}}_t^q, \end{aligned}$$

where \(a_t^q(x)\) and \(b_t^q(x)\) are functions from \({\mathbb {R}}\) to \({\mathbb {R}}\) and \({\mathbf {h}}_t^q\) are in \(\left( {\mathbf {r}}_q{\mathbb {R}}\right) ^{\perp } \) with \(\Vert {\mathbf {h}}_t^q\Vert =1\). Then by applying our decomposition to (3), we can see that

$$\begin{aligned} {\mathbf {v}}_{t+1}(x)&= \int _{-\infty }^\infty k(x-y)g(u_t(y)){\mathbf {M}}{\mathbf {v}}_t(y)\,\text {d}y \end{aligned}$$
$$\begin{aligned}&= \int _{-\infty }^\infty k(x-y)g(u_t(y)){\mathbf {M}}\left( \sum _{q=1}^{n_q}a_t^q(y){\mathbf {r}}_q + \sum _{q=1}^{n_q}b_t^q(y){\mathbf {h}}_t^q\right) \,\text {d}y \end{aligned}$$
$$\begin{aligned}&= \sum _{q=1}^{n_q}\int _{-\infty }^\infty k(x-y)g(u_t(y))a_t^q(y)\,\text {d}y {\mathbf {M}}{\mathbf {r}}_q \nonumber \\&\quad + \sum _{q=1}^{n_q}\int _{-\infty }^\infty k(x-y)g(u_t(y))b_t^q(y)\,\text {d}y {\mathbf {M}}{\mathbf {h}}_t^q. \end{aligned}$$

Since \({\mathbf {M}}{\mathbf {r}}_q ={\mathbf {r}}_q\) and \({\mathbf {M}}\) stabilizes the space \(\left( {\mathbf {r}}_q{\mathbb {R}}\right) ^{\perp } \), we can see from (49) and (52) that for all \(q \in \{1, \ldots , n_q\}\) and \(t>0\)

$$\begin{aligned} a_{t+1}^q(x)&= \int _{-\infty }^\infty k(x-y)g(u_t(y))a_t^q(y)\,\text {d}y \text { and } \end{aligned}$$
$$\begin{aligned} b_{t+1}^q(x) {\mathbf {h}}_{t+1}^q&= \int _{-\infty }^\infty k(x-y)g(u_t(y))b_t^q(y)\,\text {d}y {\mathbf {M}}{\mathbf {h}}_t^q. \end{aligned}$$

We first focus our attention on (54). By the properties of \({\mathbf {M}}\) through the matrices \({\mathbf {M}}_q\), there exists \(\delta \in (0,1)\) such that for any \(q \in \{1, \ldots , n_q\}\) and \({\mathbf {h}} \in \left( {\mathbf {r}}_q{\mathbb {R}}\right) ^{\perp }\) then \(\Vert {\mathbf {M}}{\mathbf {h}}\Vert \le \delta \Vert {\mathbf {h}}\Vert \). Thus, from (54) we can see that

$$\begin{aligned} |b_{t+1}^q(x)|&=\Vert b_{t+1}^q(x) {\mathbf {h}}_{t+1}^q\Vert \end{aligned}$$
$$\begin{aligned}&\le \int _{-\infty }^\infty k(x-y)g(u_t(y))|b_t^q(y)|\,\text {d}y \Vert {\mathbf {M}}{\mathbf {h}}_t^q\Vert \end{aligned}$$
$$\begin{aligned}&\le \delta \int _{-\infty }^\infty k(x-y)g(u_t(y))u_t(y)\frac{|b_t^q(y)|}{u_t(y)}\,\text {d}y. \end{aligned}$$

Since \(0 \le v^i_0(x) \le {u}_0(x)\) for all \(x \in {\mathbb {R}}\) for \(i=1,\ldots , n\), it is clear that \(|b_0^q(x)| \le \delta 'u_0(x)\) for all \(x\in {\mathbb {R}}\) where \(0 < \delta ' \le 1\). From iteration of (57), we have that

$$\begin{aligned} |b_{t+1}^q(x)| \le \delta ^t\delta ' \max _{x\in {\mathbb {R}}}\Vert u_0(x)\Vert . \end{aligned}$$

Since \(\delta \in (0,1)\),

$$\begin{aligned} \lim _{t\rightarrow \infty }|b_{t+1}^q(x)|&\le \lim _{t\rightarrow \infty }\delta ^t\delta ' \max _{x\in {\mathbb {R}}}\Vert u_0(x)\Vert \end{aligned}$$
$$\begin{aligned}&=0. \end{aligned}$$

Thus, \(b_{t+1}^q(x)\) converges uniformly to 0 on \({\mathbb {R}}\). Next, we turn our attention to the remaining piece of our decomposition for \(a_t^q(x)\). First, it is important to note that \(a_0^q(x)\) is a projection of \({\mathbf {v}}_0(x)\) on \({\mathbf {r}}_q\). Thus, it satisfies

$$\begin{aligned} a_0^q(x) = \frac{{\mathbf {v}}_0(x)^T\varvec{\ell }_q}{{\mathbf {r}}_q^T\varvec{\ell }_q}. \end{aligned}$$

From our assumption that the mutation class q is present at the leading edge of the front, we have

$$\begin{aligned} \frac{a_0^q(x)}{u_0(x)} \rightarrow p_0^q>0 \text { as } x\rightarrow \infty \text { and } \int _{-\infty }^\infty \text {e}^{\frac{c-\mu }{\sigma ^2}y}\left| a_0^q(y) - p_0^qu_0(y)\right| \,\text {d}y <\infty . \end{aligned}$$

Next, we consider the sequence \(|z_t(x)| = |a_t^q(x)-p_0^qu_t(x)|\) that satisfies

$$\begin{aligned} |z_{t+1}(x)| \le \int _{-\infty }^\infty k(x-y)g(u_t(y))|z_t(y)|\,\text {d}y \end{aligned}$$

with \(|z_0(x)| = |a_0^q(x)-p_0^qu_0(x)|\). By the assumption that \(0 < g(u) \le g(0)\) for all \(u\in (0,1)\), we obtain a super-solution

$$\begin{aligned} |z_{t+1}(x)| \le g(0)\int _{-\infty }^\infty k(x-y)|z_t(y)|\,\text {d}y \end{aligned}$$

with the same initial condition. The solution of (64) is bounded by the t-fold convolution

$$\begin{aligned} |z_{t}(x)| \le \left[ g(0)\right] ^t k^{*t}|z_0(x)|. \end{aligned}$$

Applying the reflected bilateral Laplace transform to (65) and using the convolution theorem, we obtain

$$\begin{aligned} {\mathcal {M}}[|z_{t}(x)|](s)&\le [g(0)]^{t}\left[ {\mathcal {M}}\left[ k(x)\right] (s)\right] ^{t}{\mathcal {M}}\left| z_0(x)|\right] (s) \end{aligned}$$
$$\begin{aligned}&= [g(0)]^{t}\left[ \text {e}^{\frac{\sigma ^2s^2}{2}+\mu s}\right] ^{t}{\mathcal {M}}\left[ |z_0(x)|\right] (s) \end{aligned}$$
$$\begin{aligned}&= [g(0)]^{t}\text {e}^{\frac{\sigma ^2ts^2}{2}+\mu ts}{\mathcal {M}}\left[ |z_0(x)|\right] (s) \end{aligned}$$
$$\begin{aligned}&= [g(0)]^{t}{\mathcal {M}}\left[ \frac{1}{\sqrt{2\pi \sigma ^2t}}\text {e}^{-\frac{(x-\mu t)^2}{2\sigma ^2t}}\right] (s){\mathcal {M}}\left[ |z_0(x)|\right] (s) \end{aligned}$$
$$\begin{aligned}&= [g(0)]^{t}{\mathcal {M}}\left[ (k_t*|z_0|)(x)\right] (s), \end{aligned}$$

where \(k_t\) is Gaussian with mean \(\mu t\) and variance \(\sigma ^2t\). Then applying the inverse transform yields

$$\begin{aligned} |z_{t}(x)|&\le [g(0)]^t(k_t*|z_0|)(x) \end{aligned}$$
$$\begin{aligned}&= [g(0)]^t\int _{-\infty }^\infty \frac{1}{\sqrt{2\pi \sigma ^2t}}\text {e}^{-\frac{(x-y-\mu t)^2}{2\sigma ^2t}}|z_0(y)|\,\text {d}y. \end{aligned}$$

In the moving half-frame with fixed \(A \in {\mathbb {R}}\), consider the element \(x_0+ct\) with \(c \ge c^* = \sqrt{2\sigma ^2\ln (g(0))} + \mu \). When we rewrite \(|z_{t}(x)|\) in this moving half-frame, we have

$$\begin{aligned} |z_{t}(x_0+ct)|&\le [g(0)]^t\int _{-\infty }^\infty \frac{1}{\sqrt{2\pi \sigma ^2t}}\text {e}^{-\frac{(x_0+ct-y-\mu t)^2}{2\sigma ^2t}}|z_{0}(y)|\,\text {d}y. \end{aligned}$$

Expanding in the exponential yields

$$\begin{aligned} \frac{(x_0+ct-y-\mu t)^2}{2\sigma ^2t}&=\frac{(x_0-y)^2}{2\sigma ^2t} + \frac{2(c-\mu )t(x_0-y)+(c-\mu )^2t^2}{2\sigma ^2t} \end{aligned}$$
$$\begin{aligned}&\ge \frac{(x_0-y)^2}{2\sigma ^2t} + \frac{c-\mu }{\sigma ^2}(x_0-y) + \ln (g(0))t. \end{aligned}$$


$$\begin{aligned} |z_{t}(x_0+ct)|&\le \frac{\text {e}^{\ln (g(0))t}}{\sqrt{2\pi \sigma ^2t}}\int _{-\infty }^\infty \text {e}^{-\frac{(x_0-y)^2}{2\sigma ^2t}}\text {e}^{-\frac{c-\mu }{\sigma ^2}(x_0-y)}\text {e}^{-\ln (g(0))t} |z_0(y)|\,\text {d}y \end{aligned}$$
$$\begin{aligned}&=\frac{1}{\sqrt{2\pi \sigma ^2t}}\int _{-\infty }^\infty \text {e}^{-\frac{(x_0-y)^2}{2\sigma ^2t}}\text {e}^{-\frac{c-\mu }{\sigma ^2}(x_0-y)}|z_0(y)|\,\text {d}y \end{aligned}$$
$$\begin{aligned}&=\frac{\text {e}^{-\frac{c-\mu }{\sigma ^2}x_0}}{\sqrt{2\pi \sigma ^2t}}\int _{-\infty }^\infty \text {e}^{-\frac{(x_0-y)^2}{2\sigma ^2t}}\text {e}^{\frac{c-\mu }{\sigma ^2}y}|z_0(y)|\,\text {d}y. \end{aligned}$$

Since \(x_0 \ge A\), we have

$$\begin{aligned} |z_{t}(x_0+ct)|&\le \frac{\text {e}^{-\frac{A(c-\mu )}{\sigma ^2}}}{\sqrt{2\pi \sigma ^2t}}\int _{-\infty }^\infty \text {e}^{\frac{c-\mu }{\sigma ^2}y}|z_0(y)|\,\text {d}y. \end{aligned}$$

From our assumption that the mutation class q is initially present at the leading edge of the front, we know that \(\int _{-\infty }^\infty \text {e}^{\frac{c-\mu }{\sigma ^2}y}|z_0(y)|\,\text {d}y<\infty \) Thus, we have that \(|z_t(x)| \rightarrow 0\) uniformly as \(t \rightarrow \infty \) in the moving half-frame \([A+ct,\infty )\) with speed \(c\ge c^*\). Returning to the definition of \(|z_t(x)|\), we can see that

$$\begin{aligned} |a_t^q(x)-p_0^qu_t(x)| \rightarrow 0 \;\text {uniformly as } t \rightarrow \infty \end{aligned}$$

in the moving half-frame \([A+ct,\infty )\). Then by putting together all the pieces, we can see from (49) that

$$\begin{aligned} \left\| {\mathbf {v}}_t(x)- \sum _{q=1}^{n_q}a_t^q(x){\mathbf {r}}_q\right\|&=\left\| \sum _{q=1}^{n_q}b_t^q(x){\mathbf {h}}_t^q\right\| \end{aligned}$$
$$\begin{aligned}&\le \sum _{q=1}^{n_q}|b_t^q(x)|\Vert {\mathbf {h}}_t^q\Vert \end{aligned}$$
$$\begin{aligned}&=\sum _{q=1}^{n_q}|b_t^q(x)|. \end{aligned}$$

Therefore, from (60) and (80) we can conclude that

$$\begin{aligned} \max _{[A+ct,\infty )}\left\| {\mathbf {v}}_t(x)- \sum _{q=1}^{n_q}p_0^qu_t(x){\mathbf {r}}_q\right\| \rightarrow 0 \text { as } t\rightarrow \infty . \end{aligned}$$

The proof of Theorem 3 is complete. \(\square \)

Proof of Theorem 4

Repeat the proof of Theorem 3 in the left moving half-frame with fixed \(A \in {\mathbb {R}}\) and consider the element \(x_0-ct\) with \(c\ge c^*_- = \sqrt{2\sigma ^2\ln (g(0))}-\mu \). From this change, the result follows in the same manner as in Theorem 3. \(\square \)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Marculis, N.G., Lewis, M.A. Inside Dynamics of Integrodifference Equations with Mutations. Bull Math Biol 82, 7 (2020).

Download citation


  • Integrodifference equations
  • Mutations
  • Neutral genetic diversity
  • Range expansion
  • Spreading speed