Predator-prey models: a review on some recent adaptations

TL;DR

The paper surveys recent predator-prey models that incorporate four biologically relevant features: the Allee effect, fear, cannibalism, and immigration, comparing how each modifies equilibria and stability and whether limit cycles arise. It leverages a range of mathematical tools, including linear stability analysis, Bendix–Dulac criteria, Hopf bifurcations, and Liènard/Kolmogorov transformations, to illuminate contexts in which these traits stabilize or destabilize dynamics. Across topics, the work highlights that the placement of the Allee effect (prey vs predator), the nature of fear, and the direction of immigration or cannibalism can dramatically alter long-term outcomes, with implications for persistence and coexistence. The review furthermore identifies gaps and open questions, offering guidance for future analytic proofs and more complex models in heterogeneous or multi-species settings.

Abstract

In the last years, predator-prey systems have increased their applications and have given rise to systems which represent more accurately different biological issues that appear in the context of interacting species. Our aim in this paper is to give a state-of-art review of recent predator-prey models which include some interesting characteristics as Allee effect, fear effect, cannibalism and immigration. We compare the qualitative results obtained for each of them, particularly regarding the equilibria, local and global stability, and the existence of limit cycles.

Figures (10)

• Figure 1: Time–population density of prey and predator for systems \ref{['sisLV']}, \ref{['LVAllee_prey']} and \ref{['AlleeD']}. The initial conditions are $x(0) = 0.1$ and $y(0) = 0.1$, and the values for the parameters are $r = 20$, $a = 15$ and $\beta= 0.6$.
• Figure 2: Behaviour of population density of prey and predator for systems \ref{['sisLV']}, \ref{['LVAllee_prey']} and \ref{['AlleeD']}. The initial conditions are $x(0) = 0.005$ and $y(0) = 0.01$, and the values for the parameters are $r = 2.5$, $a = 15$ and $\beta= 0.05$.
• Figure 3: Dynamic behaviours with $r=2$, $n=2$, $K=1$, $m=-2$, $q=1.2$, $p=0.8$ and $c=0.5$.
• Figure 4: Dynamic behaviours with $r=0.5$, $n=0.10375$, $K=1$, $m=0.001$, $q=1.2$, $p=0.175$ and $c=0.041125$.
• Figure 5: Variation of the positive equilibrium and the solutions in function of parameter $s$. The considered values of the parameters are $r = 1.6$, $d=0.8$, $a=0.2$, $p=0.3$, $c=0.8$, $n=1$ and $d_1=0.8$.