A Holling–Tanner Predator–Prey Model with Strong Allee Effect

In ecological environment, Allee effect is one of the important factors which cause significant changes to the system dynamics. In this paper, using the theory of dynamical systems, we analyze a variation of a standard cannibalistic two-dimensional prey–predator model with Holling type-II functional response in the presence of both weak and strong Allee effects. We have analyzed the impact of strong and weak Allee effects on the dynamics of a cannibalistic system, knowing the dynamics of the cannibalistic model without Allee effect. We have deduced that in the presence of cannibalism, both strong and weak Allee effects generate bistability between equilibrium points. For strong Allee effect, bistability occurs between trivial equilibrium point and predator-free equilibrium point as well as between trivial and coexistence equilibrium points. But for weak Allee effect, bistability occurs only between coexistence equilibrium points. We also pointed out that the cannibalistic system without Allee effect exhibits tristability among the trivial equilibrium point, coexistence equilibrium point having low prey concentration and coexistence equilibrium point having comparatively high prey concentration. But in the presence of strong Allee effect, cannibalistic system experiences tristability among trivial and two other stable coexistence equilibrium points. By a comprehensive bifurcation analysis, we have observed that Allee effect enriches both the local and global dynamics of the system. Here, we have reported all possible codimension-one and codimension-two bifurcations extensively by choosing cannibalism, Allee effect and predator natural death rate as the bifurcation parameters. In the analysis of bifurcations, we have explored the existence of transcritical bifurcation, saddle-node bifurcation, Hopf bifurcation, Bogdanov–Takens bifurcation and Bautin bifurcation. Our analytical findings are validated through exhaustive numerical simulations. Finally, we have reported a comparative study between the impacts of strong and weak Allee effects on the dynamics of the cannibalistic system.

The basins of attraction in a modified May–Holling–Tanner predator–prey model with Allee affect

In this paper, strong Allee effects on the bifurcation of the predator–prey model with ratio-dependent Holling type III response are considered, where the prey in the model is subject to a strong Allee effect. The existence and stability of equilibria and the detailed behavior of possible bifurcations are discussed. Specifically, the existence of saddle-node bifurcation is analyzed by using Sotomayor’s theorem, the direction of Hopf bifurcation is determined, with two bifurcation parameters, the occurrence of Bogdanov–Takens of codimension 2 is showed through calculation of the universal unfolding near the cusp. Comparing with the cases with a weak Allee effect and no Allee effect, the results show that the Allee effect plays a significant role in determining the stability and bifurcation phenomena of the model. It favors the coexistence of the predator and prey, can lead to more complex dynamical behaviors, not only the saddle-node bifurcation but also Bogdanov–Takens bifurcation. Numerical simulations and phase portraits are also given to verify our theoretical analysis.

In this paper, a Leslie–Gower model with weak Allee effect on prey and fear effect on predator is proposed. Compared with the case without fear effect on predator, the new model undergoes richer dynamic behaviors such as saddle-node bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation. Also, different from the strong Allee effect on prey, the system with weak Allee effect has bistable attractors which are a largely stable limit cycle and a stable positive equilibrium, two stable equilibria, or a stable limit cycle and a stable trivial equilibrium. When the Allee effect coefficient is intermediate, fear effect on the predator can protect the prey and the predator from being extinguished. The results in this paper can be seen as a complement to those in the literatures about the Leslie–Gower model with Allee effect and fear effect.

Dynamical complexities in the Leslie–Gower predator–prey model as consequences of the Allee effect on prey

Dynamics of a Leslie–Gower predator–prey model with Holling type II functional response, Allee effect and a generalist predator

In this work, a modified Holling–Tanner predator–prey model is analyzed, considering important aspects describing the interaction such as the predator growth function is of a logistic type; a weak Allee effect acting in the prey growth function, and the functional response is of hyperbolic type. Making a change of variables and time rescaling, we obtain a polynomial differential equations system topologically equivalent to the original one in which the non‐hyperbolic equilibrium point (0,0) is an attractor for all parameter values. An important consequence of this property is the existence of a separatrix curve dividing the behavior of trajectories in the phase plane, and the system exhibits the bistability phenomenon, because the trajectories can have different ω − limit sets; as example, the origin (0,0) or a stable limit cycle surrounding an unstable positive equilibrium point. We show that, under certain parameter conditions, a positive equilibrium may undergo saddle‐node, Hopf, and Bogdanov–Takens bifurcations; the existence of a homoclinic curve on the phase plane is also proved, which breaks in an unstable limit cycle. Some simulations to reinforce our results are also shown. Copyright © 2015 John Wiley & Sons, Ltd.

Degenerate codimension-2 cusp of limit cycles in a Holling–Tanner model with harvesting and anti-predator behavior

In this paper, a Holling–Tanner predator–prey model with generalist predators and Michaelis–Menten-type prey harvesting is investigated. We analyze the existence and stability of equilibria and find the system has at most three positive equilibria. The double positive equilibrium belongs to the cusp type, with its codimension being at least 5. We then prove that the triple positive equilibrium is either a nilpotent focus (or elliptic point) of codimension 3, or a nilpotent elliptic equilibrium with codimension no less than 4. Additionally, the system undergoes two types of bifurcations: a cusp-type degenerate Bogdanov–Takens bifurcation (codimension 3) and a Hopf bifurcation. Using numerical simulations, the system has two limit cycles, which indicates that Michaelis–Menten-type prey harvesting makes the system’s dynamics more complex.

The influences of Allee effect and density‐dependent mortality on population growth are of great significance in ecology. In this paper, we first consider a Gause‐type predator–prey model with simplified Holling type IV functional response, strong Allee effect on prey, and density‐dependent mortality of predator. It is shown that the system exhibits rich and complex dynamics like bistability, local and global bifurcations such as transcritical bifurcation, saddle‐node bifurcation, Hopf bifurcation, cusp bifurcation, Bogdanov–Takens bifurcation, Bautin bifurcation, homoclinic bifurcation, saddle‐node bifurcation of limit cycle, and heteroclinic bifurcation. Next, we separately explore the influences of Allee effect and density‐dependent mortality on the dynamics of the same model. The results show that the strong Allee effect induces the occurrence of heteroclinic bifurcation and the reduction of the number of limit cycles, while the density‐dependent mortality stabilizes the system. Since the analytical expressions of the interior equilibria are difficult to derive, the results are verified numerically.

In this paper, the dynamics of a ratio-dependent predator–prey model with strong Allee effect and Holling IV functional response is investigated by using dynamical analysis. The model is shown to have complex dynamical behaviors including subcritical or supercritical Hopf bifurcation, saddle-node bifurcation, Bogdanov–Takens bifurcation of codimension-2, a nilpotent focus or cusp of codimension-2. The codimension-2 Bogdanov–Takens bifurcation point acts as an organizing center for the whole bifurcation set. The coexistence of stable and unstable positive equilibria, homoclinic cycle is also found. Our analysis shows that the ratio-dependent model may collapse suddenly due to certain parameter variation, i.e. the numbers of predator and prey will decrease sharply to zeroes after undergoing a short time of sustained oscillations with small amplitudes. Of particular interest is that the coalescence of saddle-node bifurcation point and Hopf bifurcation point may indicate the occurrence of relaxation oscillations and the critical state of extinction of predator and prey. Numerical simulations and phase portraits including one-parameter bifurcation curve and two-parameter bifurcation curves are given to illustrate the theoretical results.

Understanding the Allee effect on endangered species is crucial for ecological conservation and management as it highly affects the extinction of a population. Due to several ecological mechanisms accounting for the Allee effect, it is necessary to study the dynamics of a predator–prey model incorporating this phenomenon. In 1999, Cosner et al. [Effects of spatial grouping on the functional response of predators, Theor Popul Biol 56:65–75, 1999] derived a new kind of functional response by considering spatially grouped predators. This paper deals with the dynamical behavior of a predator–prey system with functional response proposed by Cosner et al., and the growth of the prey population suffers a strong Allee effect. We find that the system undergoes various types of bifurcations such as Hopf bifurcation, saddle-node bifurcation, and Bogdanov–Takens bifurcation. We also observe that the model exhibits bistability and two different types of tristability phenomena. Our findings reveal that for such a kind of multistability in ecological systems, the initial population size plays a crucial role and also impacts the system’s state in the long term.

<p style='text-indent:20px;'>In this paper, we propose a Rosenzweig–MacArthur predator-prey model with strong Allee effect and trigonometric functional response. The local and global stability of equilibria is studied, and the existence of bifurcation is determined in terms of the carrying capacity of the prey, the death rate of the predator and the Allee effect. An analytic expression is employed to determine the criticality and codimension of Hopf bifurcation. The existence of supercritical Hopf bifurcation and the non-existence of Bogdanov–Takens bifurcation at the positive equilibrium are proved. A point-to-point heteroclinic cycle is also found. Biologically speaking, such a heteroclinic cycle always indicates the collapse of the system after the invasion of the predator, i.e., overexploitation occurs. It is worth pointing out that heteroclinic relaxation cycles are driven by either the strong Allee effect or the high per capita death rate. In addition, numerical simulations are given to demonstrate the theoretical results.</p>

We introduce a mathematical model to study tumor–immune interactions and analyze the effects of strong and weak Allee dynamics to capture the cooperative behavior inherent in tumor growth. In this work, we observe that the presence of Allee dynamics delays tumor proliferation at low cell densities, significantly influencing the critical thresholds that regulate tumor cell cooperation. We examine key dynamical behaviors through bifurcation analysis, including saddle-node, Hopf, Bautin, and Bogdanov–Takens (BT) bifurcations, using detailed analytical expressions and numerical continuation methods. Our model demonstrates that a strong Allee effect raises the critical threshold for tumor cell cooperation, leading to greater tumor control, as a higher antigenicity value is required to maintain equilibrium thresholds. In contrast, a weak Allee effect has a less pronounced impact on the model compared to a strong Allee effect. We also explore the case where the weak Allee effect corresponds to the parameter representing the half-saturation cancer clearance level, revealing a key dynamic behavior of the system under this condition. In this case, we prove the existence of BT bifurcation of co-dimension-three-type cusp. These findings could provide valuable insights into the transitions between elimination, equilibrium, and escape phases in the cancer immunoediting process. Moreover, our results could serve as a valuable tool for precisely calibrating these effects in the development of more effective therapeutic strategies.

Dynamics of adding variable prey refuge and an Allee effect to a predator–prey model