Allee effect and bistability in a spatially heterogeneous predator-prey model

Abstract

Abstract. A spatially heterogeneous reaction-diffusion system modelling predator-prey interaction is studied, where the interaction is governed by a Holling type II functional response. Existence of multiple positive steady states and global bifurcation branches are examined as well as related dynamical behavior. It is found that while the predator population is not far from a constant level, the prey population could be extinguished, persist or blow up depending on the initial population distributions, the various parameters in the system, and the heterogeneous environment. In particular, our results show that when the prey growth is strong, the spatial heterogeneity of the environment can play a dominant role for the presence of the Allee effect. Our mathematical analysis relies on bifurcation theory, topological methods, various comparison principles and elliptic estimates. We combine these methods with monotonicity arguments to the system through the use of some new auxiliary scalar equations, though the system itself does not keep an order structure as the competition system does. Among other things, this allows us to obtain partial descriptions of the dynamical behavior of the system.

A diffusive predator–prey model with a protection zone

“…To be more precise, when λ * > λ (small protection zone case), the dynamics of (1.2) is qualitatively similar to the case without protection zone considered in [24]: when the predator growth rate μ is small or negative, the prey-only steady state (λ, 0) is globally asymptotically stable; when μ is large, the predator-only steady state (0, μ) is globally asymptotically stable; and when μ is in the intermediate range, one or more coexistence steady states exist which attract all initial values. Allee effect or bistability could exist in the last case (see details in [24]).…”

A diffusive competition model with a protection zone

“…Similar related problems on predator-prey models were considered in [10,17,18,20,21], and [19] examined the effect of a protection zone on a diffusive predator-prey model. However, the effect of protection zones does not seem to be considered before for diffusive competition models.…”

“…It can be proved that when η * > η 0 , (1.2) has at least two positive solutions for η ∈ (η 0 , η * ). One could combine the upper and lower solution method and a bifurcation argument with η as the bifurcation parameter to prove this (see [18] for a related situation). Remark 3.17.…”

Population dispersal and Allee effect

“…For example, the baseline dynamics of population growth are governed by the Logistic equation. However, there have been abundant evidences that Allee effect plays an important role in diverse bio- [1,5,[7][8][9][10][14][15][16]18] and the references cited therein). Allee effect, defined as positive effects in the growth rate at low population density, may be strong or weak.…”