Regime shift in Rosenzweig–Macarthur predator–prey model in presence of strong Allee effect in prey

Abstract

The resilience and resistance of ecosystems to regime shifts is an emerging area of research in ecological systems. In this article, we investigate the regime shift phenomena in a Rosenzweig–MacArthur predator–prey model when a strong Allee effect is present in the prey population. Apart from strong Allee effect, we incorporate the time scale disparity between the prey and predator populations. We investigate both single-patch and two-patch models with and without dispersal delay. Our findings indicate that the Allee threshold has a negative impact on species persistence and that species extinction occurs through a regime shift due a to non-local bifurcation. This regime shift through a boundary crisis of a limit-cycle oscillator is fatal from the ecological point of view as it leads to sudden extinction. Our investigation also explores how time scale differences affect the tipping point of the regime shift event and we find that it makes species less resilient. Furthermore, we demonstrate that species dispersal in a two-patch model increases resilience and can rescue populations from extinction when a single isolated patch’s species become extinct due to a regime shift. Finally, we show that a time delay in dispersal can further enhance resistance to regime shift by large extent. The occurence of a regime shift through a boundary crisis of an oscillatory attractor has not been extensively investigated previously within the context of a two-species predator–prey model. In this study, we offer a comprehensive bifurcation analysis that delves into how species resilience can be enhanced through the effects of dispersal and time lag in dispersal. We quantify species resilience by employing basin stability metric.

Appendix A: Linear stability analysis of two patch model without dispersal delay

The two patch predator–prey model where both preys and predators migrate between the two spatially separated heterogeneous patches at a constant rate $\kappa$ is given by

$$\begin{aligned} \frac{dN_{i}}{dt}= & {} \frac{1}{\epsilon }[N_{i}(1-N_{i})(N_{i}{-}\theta _{i}){-}N_{i}P_{i}]{+}\kappa (N_{j}-N_{i}) \nonumber \\ \frac{dP_{i}}{dt}= & {} (N_{i}-\delta )P_{i}+\kappa (P_{j}-P_{i}) \nonumber \\ \end{aligned}$$

(A1)

where $i,j = 1,2$ and $i \ne j$.

This two-patch model has four equilibrium points:

• 1. (0, 0, 0, 0) known as extinction equilibrium or trivial equilibrium. It exists for all parameter values. The eigenvalues of the Jacobian matrix evaluated at this equilibrium are $\lambda _{1} = -\delta < 0$, $\lambda _{2} = -2\kappa -\delta < 0$ and the other two satisfy the equation $\lambda ^{2} + (2\kappa +\frac{\theta _{1}+\theta _{2}}{\epsilon })\lambda + (\kappa +\frac{\theta _{1}}{\epsilon })(\kappa +\frac{\theta _{2}}{\epsilon })-\kappa ^{2} = 0$. Observe that $2\kappa +\frac{\theta _{1}+\theta _{2}}{\epsilon } > 0$ and $(\kappa +\frac{\theta _{1}}{\epsilon })(\kappa +\frac{\theta _{2}}{\epsilon })-\kappa ^{2} = \frac{\kappa }{\epsilon }(\theta _{1}+\theta _{2}) + \frac{\theta _{1}\theta _{2}}{\epsilon ^{2}} > 0$. This shows that the extinction equilibrium is stable for all the parameter values.

• 2. (1, 0, 1, 0) where the prey species are at their maximum population density in both patches but the predators are extinct in both patches. This exists for all parameter values. The eigenvalues of the Jacobian matrix evaluated at this equilibrium are $\lambda _{1} = 1-\delta , \lambda _{2} = 1-2\kappa -\delta$ and the other two eigenvalues are the roots of $\lambda ^{2}+(2\kappa +\frac{2-\theta _{1}-\theta _{2}}{\epsilon })\lambda +(\kappa +\frac{1-\theta _{1}}{\epsilon })(\kappa +\frac{1-\theta _{2}}{\epsilon })-\kappa ^{2} = 0$. Obviously $2\kappa +\frac{2-\theta _{1}-\theta _{2}}{\epsilon } > 0$ and $(\kappa +\frac{1-\theta _{1}}{\epsilon })(\kappa +\frac{1-\theta _{2}}{\epsilon })-\kappa ^{2} > 0$ but $\lambda _{1} = 1-\delta >0$. Therefore, (1, 0, 1, 0) is unstable.

• 3. $(N_{10},0,N_{20},0)$ where $N_{20} = \frac{N_{10}}{\kappa \epsilon }[\kappa \epsilon - (1 - N_{10})(N_{10} - \theta _{1})]$ and $N_{10}$ is the positive real root of the equation $[\beta ^{2}-(1+\theta _{1})\beta +(\theta _{1}+\kappa \epsilon )] [\beta ^{2}-\theta _{1}\beta +\kappa \epsilon ][\beta ^{3}-(\theta _{1}+1)\beta ^{2} +(\kappa \epsilon +\theta _{1})\beta -\kappa \epsilon \theta _{2}] + \kappa ^{3}\epsilon ^{3}\beta - \kappa ^{3}\epsilon ^{3}\theta _{1} = 0$. This equilibrium exists if $\kappa \epsilon > (1 - N_{10})(N_{10} - \theta _{1})$ so that $N_{20}>0$. The eigenvalues are the roots of the characteristic equation given by $[ (\lambda +\delta +\kappa -N_{10})(\lambda +\delta +\kappa -N_{20}) -\kappa ^{2}][(\lambda -M_{1}^{*})(\lambda -L_{1}^{*})-\kappa ^{2}] = 0$, where $M_{1}^{*} = \frac{1}{\epsilon }[(1 - N_{10})(N_{10} - \theta _{1}) + N_{10}(1 - 2N_{10} + \theta _{1})] - \kappa$ and $L_{1}^{*} = \frac{1}{\epsilon }[(1 - N_{20})(N_{20} - \theta _{2}) + N_{20}(1 - 2N_{20} + \theta _{2})] - \kappa$. This fixed point is stable only if

  • $2(\delta +\kappa ) > N_{10}+N_{20}$

  • $(\delta -N_{10})(\delta -N_{20}) > \kappa (N_{10}+N_{20}-2\delta )$

  • $M_{1}^{*}+L_{1}^{*} < 0$

  • $M_{1}^{*}L_{1}^{*} > \kappa ^{2}$

• 4. The coexistence equilibrium $(N_{100}, P_{100}, N_{200}, P_{200})$ where all four populations survive. The expression for the coexistence equilibrium point is quite complex, we have chosen to not extensively discuss its stability in this context.