Global stability in a modified Leslie-Gower type predation model assuming mutual interference among generalist predators.

In the ecological literature, mutual interference (self-interference) or competition among predators (CAP) to effect the harvesting of their prey has been modeled through different mathematical formulations. In this work, the dynamical properties of a Leslie-Gower type predation model is analyzed, incorporating one of these forms, which is described by the function $g\left(y\right) =y^{\beta }$, with $0<\beta <1$. This function $g$ is not differentiable for $y=0$, and neither the Jacobian matrix of the system is not defined in the equilibrium points over the horizontal axis ($x-axis$). To determine the nature of these points, we had to use a non-standard methodology. Previously, we have shown the fundamental properties of the Leslie-Gower type model with generalist predators, to carry out an adequate comparative analysis with the model where the competition among predators (CAP) is incorporated. The main obtained outcomes in both systems are: (i) The unique positive equilibrium point, when exists, is globally asymptotically stable (g.a.s), which is proven using a suitable Lyapunov function. (ii) There not exist periodic orbits, which was proved constructing an adequate Dulac function.

Dynamics of a modified Leslie-Gower predation model considering a generalist predator and the hyperbolic functional response.

In the ecological literature,many models for the predator-prey interactions have been well formulated but partially analyzed.Assuming this analysis to be true and complete,some authors use that results to study a more complex relationship among species (food webs).Others employ more sophisticated mathematical tools for the analysis,without further questioning.The aim of this paper is to extend,complement and enhance the results established in an earlier article referred to a modified Leslie-Gower model.In that work,the authors proved only the boundedness of solutions,the existence of an attracting set,and the global stability of a single equilibrium point at the interior of the first quadrant.In this paper,new results for the same model are proven,establishing conditions in the parameter space for which up two positive equilibria exist.Assuming there exists a unique positive equilibrium point,we have proved,the existence of:i) a separatrix curve Σ,dividing the trajectories in the phase plane,which can have different ω-limit,ii) a subset of the parameter space in which two concentric limit cycles exist,the innermost unstable and the outermost stable.Then,there exists the phenomenon of tri-stability,because simultaneously,it has:a local stable positive equilibrium point, a stable limit cycle,and an attractor equilibrium point over the vertical axis.Therefore,we warn the model studied have more rich and interesting properties that those shown that earlier papers.Numerical simulations and a bifurcation diagram are given to endorse the analytical results.

Bifurcations and multistability on the May-Holling-Tanner predation model considering alternative food for the predators.

In this paper a modified May-Holling-Tanner predator-prey model is analyzed, considering an alternative food for predators, when the quantity of prey i scarce. Our obtained results not only extend but also complement existing ones for this model, achieved in previous articles. The model presents rich dynamics for different sets of the parameter values; it is possible to prove the existence of: (i) a separatrix curve on the phase plane dividing the behavior of the trajectories, which can have different ω-limit; this implies that solutions nearest to that separatrix are highly sensitive to initial conditions, (ii) a homoclinic curve generated by the stable and unstable manifolds of a saddle point in the interior of the first quadrant, whose break generates a non-infinitesimal limit cycle, (iii) different kinds of bifurcations, such as: saddle-node, Hopf, Bogdanov-Takens, homoclinic and multiple Hopf bifurcations. (iv) up to two limit cycles surrounding a positive equilibrium point, which is locally asymptotically stable. Thus, the phenomenon of tri-stability can exist, since simultaneously can coexist a stable limit cycle, joint with two locally asymptotically stable equilibrium points, one of them over the y-axis and the other positive singularity. Numerical simulations supporting the main mathematical outcomes are shown and some of their ecological meanings are discussed.

Modeling a SI epidemic with stochastic transmission: hyperbolic incidence rate.

In this paper a stochastic susceptible-infectious (SI) epidemic model is analysed, which is based on the model proposed by Roberts and Saha (Appl Math Lett 12: 37-41, 1999), considering a hyperbolic type nonlinear incidence rate. Assuming the proportion of infected population varies with time, our new model is described by an ordinary differential equation, which is analogous to the equation that describes the double Allee effect. The limit of the solution of this equation (deterministic model) is found when time tends to infinity. Then, the asymptotic behaviour of a stochastic fluctuation due to the environmental variation in the coefficient of disease transmission is studied. Thus a stochastic differential equation (SDE) is obtained and the existence of a unique solution is proved. Moreover, the SDE is analysed through the associated Fokker-Planck equation to obtain the invariant measure when the proportion of the infected population reaches steady state. An explicit expression for invariant measure is found and we study some of its properties. The long time behaviour of deterministic and stochastic models are compared by simulations. According to our knowledge this incidence rate has not been previously used for this type of epidemic models.

Uniqueness of limit cycles and multiple attractors in a Gause-type predator-prey model with nonmonotonic functional response and Allee effect on prey.

The main purpose of this work is to analyze a Gause type predator-prey model in which two ecological phenomena are considered: the Allee effect affecting the prey growth function and the formation of group defence by prey in order to avoid the predation. We prove the existence of a separatrix curves in the phase plane, determined by the stable manifold of the equilibrium point associated to the Allee effect, implying that the solutions are highly sensitive to the initial conditions. Trajectories starting at one side of this separatrix curve have the equilibrium point (0,0) as their ω-limit, while trajectories starting at the other side will approach to one of the following three attractors: a stable limit cycle, a stable coexistence point or the stable equilibrium point (K,0) in which the predators disappear and prey attains their carrying capacity. We obtain conditions on the parameter values for the existence of one or two positive hyperbolic equilibrium points and the existence of a limit cycle surrounding one of them. Both ecological processes under study, namely the nonmonotonic functional response and the Allee effect on prey, exert a strong influence on the system dynamics, resulting in multiple domains of attraction. Using Liapunov quantities we demonstrate the uniqueness of limit cycle, which constitutes one of the main differences with the model where the Allee effect is not considered. Computer simulations are also given in support of the conclusions.

Multiple limit cycles in a Gause type predator-prey model with Holling type III functional response and Allee effect on prey.

This work aims to examine the global behavior of a Gause type predator-prey model considering two aspects: (i) the functional response is Holling type III and, (ii) the prey growth is affected by the Allee effect. We prove the origin of the system is an attractor equilibrium point for all parameter values. It has also been shown that it is the ω-limit of a wide set of trajectories of the system, due to the existence of a separatrix curve determined by the stable manifold of the equilibrium point (m,0), which is associated to the Allee effect on prey. When a weak Allee effect on the prey is assumed, an important result is obtained, involving the existence of two limit cycles surrounding a unique positive equilibrium point: the innermost cycle is unstable and the outermost stable. This property, not yet reported in models considering a sigmoid functional response, is an important aspect for ecologists to acknowledge as regards the kind of tristability shown here: (1) the origin; (2) an interior equilibrium; and (3) a limit cycle of large amplitude. These models have undoubtedly been rather sensitive to disturbances and require careful management in applied conservation and renewable resource contexts.

Population-level consequences of antipredator behavior: a metaphysiological model based on the functional ecology of the leaf-eared mouse.

We present a predator-prey metaphysiological model, based on the available behavioral and physiological information of the sigmodontine rodent Phyllotis darwini. The model is focused on the population-level consequences of the antipredator behavior, performed by the rodent population, which is assumed to be an inducible response of predation avoidance. The decrease in vulnerability is explicitly considered to have two associated costs: a decreasing foraging success and an increasing metabolic loss. The model analysis was carried out on a reduced form of the system by means of numerical and analytical tools. We evaluated the stability properties of equilibrium points in the phase plane, and carried out bifurcation analyses of rodent equilibrium density under varying conditions of three relevant parameters. The bifurcation parameters chosen represent predator avoidance effectiveness (A), foraging cost of antipredator behavior (C(1)'), and activity-metabolism cost (C(4)'). Our analysis suggests that the trade-offs involved in antipredator behavior plays a fundamental role in the stability properties of the system. Under conditions of high foraging cost, stability decreases as antipredator effectiveness increases. Under the complementary scenario (not considering the highest foraging costs), the equilibria are either stable when both costs are low, or unstable when both costs are higher, independent of antipredator effectiveness. No evidence of stabilizing effects of antipredator behavior was found.