Multiple hydra effect in a predator–prey model with Allee effect and mutual interference in the predator

Stochastic sensitivity analysis and early warning signals of critical transitions in a tri-stable prey-predator system with noise

Near a tipping point, small changes in a certain parameter cause an irreversible shift in the behavior of a system, called critical transitions. Critical transitions can be observed in a variety of complex dynamical systems, ranging from ecology to financial markets, climate change, molecular bio-systems, health, and disease. As critical transitions can occur suddenly and are hard to manage, it is important to predict their occurrence. Although it is very tough to predict such critical transitions, various recent works suggest that generic early warning signals can detect the situation when systems approach a critical point. The most important indicator that predicts the risk of an upcoming critical transition is critical slowing down (CSD). CSD indicates a slow recovery rate from external perturbations of the stable state close to a bifurcation point. In this contribution, we study a two dimensional prey-predator model. Without any noise, the prey-predator model shows bistability and tri-stability due to the Allee effect in predators. We explore the critical transitions when external noise is added to the prey-predator system. We investigate early warning indicators, e.g., recovery rate, lag-1 autocorrelation, variance, and skewness to predict the critical transition. We explore the confidence domain method using the stochastic sensitivity function (SSF) technique near a stable equilibrium point to find a threshold value of noise intensity for a transition. The SSF technique in a two stage transition through confidence ellipse is described. We also show that the possibility of a transition to the predator-free state is independent of initial conditions. Our result may serve as a paradigm to understand and predict the critical transition in a two dimensional system.

Bifurcation analysis of the predator-prey model with the Allee effect in the predator

The use of predator–prey models in theoretical ecology has a long history, and the model equations have largely evolved since the original Lotka–Volterra system towards more realistic descriptions of the processes of predation, reproduction and mortality. One important aspect is the recognition of the fact that the growth of a population can be subject to an Allee effect, where the per capita growth rate increases with the population density. Including an Allee effect has been shown to fundamentally change predator–prey dynamics and strongly impact species persistence, but previous studies mostly focused on scenarios of an Allee effect in the prey population. Here we explore a predator–prey model with an ecologically important case of the Allee effect in the predator population where it occurs in the numerical response of predator without affecting its functional response. Biologically, this can result from various scenarios such as a lack of mating partners, sperm limitation and cooperative breeding mechanisms, among others. Unlike previous studies, we consider here a generic mathematical formulation of the Allee effect without specifying a concrete parameterisation of the functional form, and analyse the possible local bifurcations in the system. Further, we explore the global bifurcation structure of the model and its possible dynamical regimes for three different concrete parameterisations of the Allee effect. The model possesses a complex bifurcation structure: there can be multiple coexistence states including two stable limit cycles. Inclusion of the Allee effect in the predator generally has a destabilising effect on the coexistence equilibrium. We also show that regardless of the parametrisation of the Allee effect, enrichment of the environment will eventually result in extinction of the predator population.

Bifurcations and hydra effects in Bazykin's predator-prey model

We consider Bazykin's model to address harvesting induced stability exchanges through bifurcation analysis. We examine the existence of hydra effects and analyze the stock pattern under predator harvesting. Prey harvesting cannot produce hydra effects in our model, whereas predator harvesting may cause multiple hydra effects. Our study reveals that type II response function and mutual interference among predators jointly induce multiple hydra effects and bistability. Bifurcations such as single Hopf-bifurcation, multiple Hopf-bifurcations and multiple saddle-node bifurcations appear for increasing harvesting rate on the predators. However, over-exploitation of the predators cannot generate any such bifurcation in our study. In simulations, the maximum sustainable yield (MSY) exists at a globally stable state. When predator is culled under increasing effort, basin of attraction of the equilibrium corresponding to the higher predator stock gets expanded, which alternatively is in favor of stock benefit for predators. The ecological theory developed in this study might be useful to understand conservation policy and fishery management.

Stability switching and hydra effect in a predator-prey metapopulation model

A metapopulation model is investigated to explore how the spatial heterogeneity affects predator-prey interactions. A Rosenzweig-MacArthur (RM) predator-prey model with dispersal of both the prey and predator is formulated. We propose such a system as a well mixed spatial model. Here, partially mixed spatial models are defined in which the dispersal of only one of the communities (prey or predator) is considered. In our study, the spatial heterogeneity is induced by dissimilar (unbalanced) dispersal rates between the patches. A large difference between the predator dispersal rates may stabilize the unstable positive equilibrium of the model. The existence of two ecological phenomena are found under independent harvesting strategy: stability switching and hydra effect. When prey or predator is harvested in a heterogenious environment, a positive stable steady state becomes unstable with increasing the harvesting effort, and a further increase in the effort leads to a stable equilibrium. Thus, a stability switching happens. Furthermore, the predator biomass (at stable state) in both the patches (and hence total predator stock) increases when the patch with a higher predator density is harvested; resulting a hydra effect. These two phenomena do not occur in the non-spatial RM model. Hence, spatial heterogeneity induces stability switching and hydra effect.

The interplay between fishery yield and top predator culling in a multispecies fishery context

In food webs, fishery can play the role of top predator, competing thus with other top predators for valuable food resources. In this view, it has been claimed in fisheries management that culling of top predators can be a means to improve fishery yield. To investigate this hypothesis, we use theoretical population models to assess in a multispecies context how fishery yield from target species harvest responds to top predator cull. Defying crisp summary, the four analysed food web models show that this response may be either positive or negative or both, indicating that in terms of multispecies fishery management the harvest yield may not accrue as a consequence of predator removal. In addition, this multitude of behaviours points also to the fact that the response of fishery yield to top predator cull may be difficult to assess.

IPM strategies to a discrete switching predator-prey model induced by a mate-finding Allee effect

This paper proposes a discrete switching predator-prey model with a mate-finding Allee effect, where also switches are guided by Allee effect. One of the strategies analysed is to use a chemical in order to prevent the pest outbreak when the pest population is free of Allee effect. In this paper, we first study analytically the dynamic behaviors of the two subsystems and the equilibria and their stability of the switched system. Then we provide numerical bifurcation analyses for the switched discrete system. These show that the switched discrete system may have very complex dynamics by 2-parameter bifurcation diagrams which divide the space into regions and study equilibria, and 1-dimensional bifurcation diagrams which reveal that the system has periodic, chaotic solutions, period doubling bifurcations and so on. Furthermore, we try to refer the key parameters and initial densities of both populations associated with pest outbreaks and study their biological implications.

Harvesting induced stability and instability in a tri-trophic food chain

Non-equilibrium dynamics in the form of oscillations or chaos is often found to be a natural phenomenon in complex ecological systems. In this paper, we first analyze a tri-trophic food chain, which is an extension of the Rosenzweig-MacArthur di-trophic food chain. We then explore the impact of harvesting individual trophic levels to answer the following questions : a) when a non-equilibrium dynamics persists, b) whether it can locally be stabilized to a steady state, c) when the system switches from a stable steady state to a non-equilibrium dynamics and d) whether the Maximum Sustainable Yield (MSY) always exists when the top predator is harvested. It is shown that searching for a general theory to unify the harvesting induced stability must take into account the number of trophic levels and the degree of species enrichment, the outcomes that cannot be obtained from the earlier reports on prey-predator models. We also identify the situation where harvesting induces instability switching: the non-equilibrium state enters into a stable steady-state and then, upon more intensive harvesting, the steady-state again loses its stability. One of the new and important results is also that the MSY may not exist for harvesting the top predator. In general, our results contribute to biological conservation theory, fishery and ecosystem biodiversity management.