An Evolutionary Beverton-Holt Model

Abstract

The classic Beverton-Holt (discrete logistic) difference equation, which arises in population dynamics, has a globally asymptotically stable equilibrium (for positive initial conditions) if its coefficients are constants. If the coefficients change in time, then the equation becomes nonautonomous and the asymptotic dynamics might not be as simple. One reason the coefficients can change in time is their evolution by natural selection. If the model coefficients are functions of a heritable phenotypic trait subject to natural selection then, by standard methods for modeling evolution, the model becomes a planar system of coupled difference equations, consisting of a Beverton-Holt type equation for the population dynamics and a difference equation for the dynamics of the mean phenotypic trait. We consider a case when the trait equation uncouples from the population dynamic equation and obtain criteria under which the evolutionary system has globally asymptotically stable equilibria or periodic solutions.

Appendix

Lemma 1

Consider the nonautonomous linear difference equation

$$\begin{aligned} y_{t+1}=\alpha _{t}y_{t}+\beta _{t} \end{aligned}$$

(21)

for $t=0,1,2,\ldots .$ Assume $\alpha _{t},\beta _{t}\ge 0$ and

$$ \lim _{t\rightarrow +\infty }\alpha _{t}=\alpha ^{e},\quad \lim _{t\rightarrow +\infty }\beta _{t}=\beta ^{e}. $$

If $\alpha ^{e}<1$ then for all $y_{0}>0$

$$ \lim _{t\rightarrow +\infty }y_{t}=y^{e}:=\frac{\beta ^{e}}{1-\alpha ^{e} }. $$

Proof

If we define $w_{t}:= y_{t}-y_{e}$, then

$$ w_{t+1}=\alpha _{t}w_{t}+q_{t} $$

where

$$ q_{t}:=\left( \alpha _{t}-\alpha ^{e}\right) y^{e}+\beta _{t}-\beta ^{e} $$

and hence

$$ \lim _{t\rightarrow +\infty }q_{t}=0. $$

By induction

$$ w_{t+1}=\left( {\displaystyle \prod \limits _{i=0}^{t}} \alpha _{i}\right) w_{0}+ {\displaystyle \sum \limits _{j=1}^{t}} \left( {\displaystyle \prod \limits _{i=j}^{t}} \alpha _{i}\right) q_{j-1}+q_{t}. $$

Let $\bar{\alpha }\$and $\bar{q}>0$ be upper bounds for the bounded sequences $\alpha _{t}$ and $\left| q_{t}\right|$ respectively. Choose a positive $\rho <1$. Since $\alpha ^{e}<1$ and $\lim _{t\rightarrow +\infty } b_{t}=b^{e}$, for arbitrary $\varepsilon >0$, we can find a $T>0$ such that $t\ge T$ implies

$$ 0\le \alpha _{t}\le \rho ,\quad \left| q_{t}\right| \le \varepsilon \left( 1-\rho \right) . $$

For $t\ge T$ we have

$$\begin{aligned} \left| w_{t+1}\right|&\le \left( {\displaystyle \prod \limits _{i=0}^{T}} \alpha _{i}\right) \left( {\displaystyle \prod \limits _{i=T+1}^{t}} \alpha _{i}\right) \left| w_{0}\right| + {\displaystyle \sum \limits _{j=1}^{T}} \left( {\displaystyle \prod \limits _{i=j}^{t}} \alpha _{i}\right) \left| q_{j-1}\right| \\&\qquad + {\displaystyle \sum \limits _{j=T+1}^{t}}\left( {\displaystyle \prod \limits _{i=j}^{t}} \alpha _{i}\right) \left| q_{j-1}\right| +\left| q_{t}\right| \\&\le \bar{\alpha }^{T+1}\rho ^{t-T}\left| w_{0}\right| + {\displaystyle \sum \limits _{j=1}^{T}} \left( {\displaystyle \prod \limits _{i=j}^{T}} \alpha _{i}\right) \left( {\displaystyle \prod \limits _{i=T+1}^{t}} \alpha _{i}\right) \left| q_{j-1}\right| \\&\qquad + {\displaystyle \sum \limits _{j=T+1}^{t}} \rho ^{t-j+1}\varepsilon \left( 1-\rho \right) +\varepsilon \left( 1-\rho \right) \end{aligned}$$

$$\begin{aligned} \left| w_{t+1}\right|&\le \bar{\alpha }^{T+1}\rho ^{t-T}\left| w_{0}\right| +\rho ^{t-T} \bar{q} {\displaystyle \sum \limits _{j=1}^{T}} \bar{\alpha }^{T-j+a}+\frac{1}{1-\rho }\varepsilon \left( 1-\rho \right) . \end{aligned}$$

Letting $t\rightarrow +\infty$ we obtain

$$ \lim _{t\rightarrow +\infty }\sup \left| w_{t+1}\right| \le \varepsilon . $$

Because $\varepsilon >0$ is arbitrary, we conclude $\lim _{t\rightarrow +\infty }\left| w_{t+1}\right| =0$.