Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Population size in stochastic multi-patch ecological models

Alexandru Hening, Siddharth Sabharwal

TL;DR

The paper addresses how dispersal and environmental stochasticity interact to shape the total population size in $n$-patch ecological models, focusing on Beverton–Holt and Hassell functional responses. It adopts stochastic persistence theory and linearization around zero to obtain persistence/extinction criteria via the metapopulation growth rate $M$, and uses Cuello's small-noise expansions to quantify shifts in the stationary total population relative to deterministic baselines. The work derives explicit approximations in slow and fast dispersal regimes, analyzes Markovian environmental forcing, and provides detailed small-noise expansions for a two-patch Beverton–Holt system, revealing that the sign and magnitude of parameter covariances crucially determine when environmental fluctuations increase or decrease the total population. Simulations corroborate the theoretical predictions, showing that dispersal can both rescue populations from extinction and, depending on the functional response and noise structure, either amplify or diminish the total population at stationarity. The results offer insights into conservation strategies under environmental variability and highlight avenues for extending the analysis to continuous-time dynamics.

Abstract

We look at the interaction of dispersal and environmental stochasticity in $n$-patch models. We are able to prove persistence and extinction results even in the setting when the dispersal rates are stochastic. As applications we look at Beverton-Holt and Hassell functional responses. We find explicit approximations for the total population size at stationarity when we look at slow and fast dispersal. In particular, we show that if dispersal is small then in the Beverton-Holt setting, if the carrying capacity is random, then environmental fluctuations are always detrimental and decrease the total population size. Instead, in the Hassell setting, if the inverse of the carrying capacity is made random, then environmental fluctuations always increase the population size. Fast dispersal can save populations from extinction and therefore increase the total population size. We also analyze a different type of environmental fluctuation which comes from switching environmental states according to a Markov chain and find explicit approximations when the switching is either fast or slow - in examples we are able to show that slow switching leads to a higher population size than fast switching. Using and modifying some approximation results due to Cuello, we find expressions for the total population size in the $n=2$ patch setting when the growth rates, carrying capacities, and dispersal rates are influenced by random fluctuations. We find that there is a complicated interaction between the various terms and that the covariances between the various random parameters (growth rate, carrying capacity, dispersal rate) play a key role in whether we get an increase or a decrease in the total population size. Environmental fluctuations turn to sometimes be beneficial -- this show that not only dispersal, but also environmental stochasticity can lead to an increase in population size.

Population size in stochastic multi-patch ecological models

TL;DR

The paper addresses how dispersal and environmental stochasticity interact to shape the total population size in -patch ecological models, focusing on Beverton–Holt and Hassell functional responses. It adopts stochastic persistence theory and linearization around zero to obtain persistence/extinction criteria via the metapopulation growth rate , and uses Cuello's small-noise expansions to quantify shifts in the stationary total population relative to deterministic baselines. The work derives explicit approximations in slow and fast dispersal regimes, analyzes Markovian environmental forcing, and provides detailed small-noise expansions for a two-patch Beverton–Holt system, revealing that the sign and magnitude of parameter covariances crucially determine when environmental fluctuations increase or decrease the total population. Simulations corroborate the theoretical predictions, showing that dispersal can both rescue populations from extinction and, depending on the functional response and noise structure, either amplify or diminish the total population at stationarity. The results offer insights into conservation strategies under environmental variability and highlight avenues for extending the analysis to continuous-time dynamics.

Abstract

We look at the interaction of dispersal and environmental stochasticity in -patch models. We are able to prove persistence and extinction results even in the setting when the dispersal rates are stochastic. As applications we look at Beverton-Holt and Hassell functional responses. We find explicit approximations for the total population size at stationarity when we look at slow and fast dispersal. In particular, we show that if dispersal is small then in the Beverton-Holt setting, if the carrying capacity is random, then environmental fluctuations are always detrimental and decrease the total population size. Instead, in the Hassell setting, if the inverse of the carrying capacity is made random, then environmental fluctuations always increase the population size. Fast dispersal can save populations from extinction and therefore increase the total population size. We also analyze a different type of environmental fluctuation which comes from switching environmental states according to a Markov chain and find explicit approximations when the switching is either fast or slow - in examples we are able to show that slow switching leads to a higher population size than fast switching. Using and modifying some approximation results due to Cuello, we find expressions for the total population size in the patch setting when the growth rates, carrying capacities, and dispersal rates are influenced by random fluctuations. We find that there is a complicated interaction between the various terms and that the covariances between the various random parameters (growth rate, carrying capacity, dispersal rate) play a key role in whether we get an increase or a decrease in the total population size. Environmental fluctuations turn to sometimes be beneficial -- this show that not only dispersal, but also environmental stochasticity can lead to an increase in population size.

Paper Structure

This paper contains 23 sections, 10 theorems, 171 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The interaction of dispersal and stochasticity
  3. Deterministic $n$ patch model
  4. Stochastic $n$-patch model
  5. Stochastic $n$-patch Beverton Holt model
  6. Slow and fast dispersal in the Beverton-Holt $n$ patch model
  7. Two-patch model
  8. Hassell functional response
  9. Slow dispersal in the Hassell $n$ patch model
  10. Markovian environmental forcing
  11. Small noise approximations
  12. Small noise approximations for a Beverton-Holt $2$-patch model
  13. Heuristic proof for small noise and small dispersal
  14. Simulations
  15. Simulations
  16. ...and 8 more sections

Key Result

Theorem 2.1

(kirkland2006evolution) Assume that The following classification of the dynamics holds:

Figures (3)

  • Figure 1: Total population size versus dispersal rate ($\delta$) in the two-patch Beverton-Holt model, with $c=0.75$. Consider random variables $r_1(t),r_2(t)$ to be uncorrelated: $\Sigma=sI$. A: $r_1=3$, $K_1=2$ and $r_2=1.5$, $K_2=1.5$. B: $r_1=3.2$, $K_1=3.85$ and $r_2=1.5$, $K_2=1.37$. C: $r_1=3.4$, $K_1=8.4$ and $r_2=1.5$, $K_2=1.37$. D: $r_1=2$, $K_1=1$ and $r_2=1.25$, $K_2=1.25$. The red curve corresponds to $s=5\times 10^{-4}$, yellow to $s=0.25$, and the purple one to $s=1.5$. In all four of these plots we observe that adding noise does not seem to change the qualitative relationship between total population and dispersal rate. In the monotonically detrimental case (D), higher noise seems to temper the effect of dispersal, even though it remains monotonically detrimental.
  • Figure 2: Total population size versus dispersal rate ($\delta$) in the 2D Beverton-Holt model. We assume the random variables $K_1(t),K_2(t)$ to be uncorrelated: $Cov((K_1(t),K_2(t)))=sI$. The model parameters $r_1,r_2,K_1,K_2$ are the same as in Figure \ref{['f:BH6']}. The blue curves in all plots depict the deterministic population. A: the red curve corresponds to $s=5\times 10^{-3}$, the yellow curve to $s=0.02$, and the purple one to $s=0.05$. B the red curve corresponds to $s=5\times 10^{-3}$, the yellow one to $s=0.02$, and the purple one to $s=0.05$ . C: the red curve corresponds to $s=0.05$, the yellow to $s=0.25$, and the purple $s=0.85$ . D: the red curve corresponds to $s=5\times 10^{-3}$, the yellow to $s=0.05$, and the purple $s=0.1$. We observe that in all four cases, adding noise decreases total population while preserving the qualitative relationship between total population and dispersal rate ($\delta$).
  • Figure 3: Total population size versus dispersal rate ($\delta$) in the 2D Hassell model. We assume the random variables $K_1(t),K_2(t)$ to be uncorrelated: $Cov((K_1(t),K_2(t)))=sI$. The blue curves in all plots depict the deterministic population. A: $\alpha_1=2.25$, $K_1=0.025$ and $\alpha_2=1.862$$K_2=0.0185$. The red curve corresponds to $s=1\times 10^{-5}$, yellow to $s=1\times 10^{-4}$, and purple $s=5\times 10^{-4}$. B: $\alpha_1=2.25$, $K_1=0.025$ and $\alpha_2=1.75$, $K_2=0.0195$. The red curve corresponds to $s=5\times 10^{-5}$, yellow to $s=5\times 10^{-4}$, and purple $s=5\times 10^{-3}$. C: $\alpha_1=2.25$, $K_1=0.025$ and $\alpha_2=1.5$, $K_2=0.0225$. the red curve corresponds to $s=5\times 10^{-4}$, yellow to $s=5\times 10^{-3}$, and purple $s=0.05$ . D: $\alpha_1=1.783$, $K_1=0.0588$ and $\alpha_2=1.838$, $K_2=0.0725$. The red curve corresponds to $s=1\times 10^{-4}$, yellow to $s=5\times 10^{-4}$, and purple $s=5\times 10^{-3}$. We observe that adding small noise does not affect the total population by much, however if the noise is significant we see an increase in total population in panels B, C and D.

Theorems & Definitions (28)

  • Theorem 2.1
  • Remark 2.1
  • Remark 2.2
  • Theorem 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Theorem 2.3
  • Remark 2.6
  • Remark 2.7
  • ...and 18 more