Predator Population Dynamics Involving Exponential Integral Function When Prey Follows Gompertz Model

The current study investigates the predator-prey problem with assumptions that interaction of predation has a little or no effect on prey population growth and the prey’s grow rate is time dependent. The prey is assumed to follow the Gompertz growth model and the respective predator growth function is constructed by solving ordinary differential equations. The results show that the predator population model is found to be a function of the well known exponential integral function. The solution is also given in Taylor’s series. Simulation study shows that the predator population size eventually converges either to a finite positive limit or zero or diverges to positive infinity. Under certain conditions, the predator population converges to the asymptotic limit of the prey model. More results are included in the paper.

Exploring the dynamics of a Holling-Tanner model with cannibalism in both predator and prey population

Cannibalism is an intriguing life history trait, that has been considered primarily in the predator, in predator-prey population models. Recent experimental evidence shows that prey cannibalism can have a significant impact on predator-prey population dyna- mics in natural communities. Motivated by these experimental results, we investigate a ratio-dependent Holling-Tanner model, where cannibalism occurs simultaneously in both the predator and prey species. We show that depending on parameters, whilst prey or predator cannibalism acting alone leads to instability, their joint effect can actually stabilize the unstable interior equilibrium. Furthermore, in the spatially explicit model, we find that depending on parameters, prey and predator cannibalism acting jointly can cause spatial patterns to form, while not so acting individually. We discuss ecologicalconsequences of these findings in light of food chain dynamics, invasive species control and climate change.

Global stability of a stochastic predator-prey model with Allee effect

In this paper, we study a stochastic predator-prey model with Beddington-DeAngelis functional response and Allee effect, and show that there is a unique global positive solution to the system with the positive initial value. Sufficient conditions for global asymptotic stability are established. Some simulation figures are introduced to support the analytical findings.

ANALYSIS OF A PREDATOR-PREY MODEL WITH DISEASE IN THE PREY

In this paper, we discuss the behavior of a predator-prey model with disease in the prey with and without stochastic perturbation, respectively. First, we briefly give the dynamic of the deterministic system, by analyzing stabilities of its four equilibria. Then, we consider the asymptotic behavior of the stochastic system. By Lyapunov analysis methods, we show the stochastic stability and its long time behavior around the equi- librium of the deterministic system. We obtain there are similar properties between the stochastic system and its corresponding deterministic system, when white noise is small. But large white noise can make a unstable deterministic system to be stable.

DYNAMICS OF A DIFFUSIVE PREDATOR-PREY MODEL WITH ADDITIVE ALLEE EFFECT

In this paper, we investigate the dynamics of a diffusive predator prey model with Holling-II functional response and the additive Allee effect in prey. We show the local and global asymptotical stability of the positive equilibrium, and give the conditions of the existence of the Hopf bifurcation. By carrying out global qualitative and bifurcation analysis, it is shown that the weak and strong Allee effects in prey can induce different dynamical behavior in the predator-prey model. Furthermore, we use some numerical simulations to illustrate the dynamics of the model. The results may be helpful for controlling and managing the predator-prey system.

Traveling wave solutions for a diffusive predator-prey model with predator saturation and competition

The purpose of this paper is to study the traveling wave solutions of a diffusive predator- prey model with predator saturation and competition functional response. The system admits three equilibria： a zero equilibrium E0, a boundary equilibrium E1 and a posi- tive equilibrium E. under some conditions. We establish the existence of two types of traveling wave solutions which connect E0 and E. and E1 and E., respectively. Our main arguments are based on a simplified shooting method, a sandwich method and constructions of appropriate Lyapunov functions. Our particular interest is to investi- gate the oscillation of both types of traveling wave solutions when they approach the positive equilibrium.

BIFURCATION CONTROL OF A PREDATOR-PREY MODEL BASED ON NUTRITION KINETICS

The existence conditions of Hopf bifurcation for a predator prey model based on nutri- tion kinetics are given. The two results may appear as follows： one is that the model has a stable periodic trajectory from Hopf bifurcation, which shows the system is in an eco- logical balance; the other is that periodic trajectory from Hopf bifurcation is unstable, which indicates the system is in a sharp or catastrophic loss of stability. For the latter, a bifurcation controller is designed to make the periodic trajectory stable. Finally, some simulations are carried out to prove the results.

On Nonlinear Conformable Fractional Order Dynamical System via Differential Transform Method

This article studies a nonlinear fractional order Lotka-Volterra prey-predator type dynamical system.For the proposed study,we consider the model under the conformable fractional order derivative(CFOD).We investigate the mentioned dynamical system for the existence and uniqueness of at least one solution.Indeed,Schauder and Banach fixed point theorems are utilized to prove our claim.Further,an algorithm for the approximate analytical solution to the proposed problem has been established.In this regard,the conformable fractional differential transform(CFDT)technique is used to compute the required results in the form of a series.Using Matlab-16,we simulate the series solution to illustrate our results graphically.Finally,a comparison of our solution to that obtained for the Caputo fractional order derivative via the perturbation method is given.

Impact of predation in the spread of an infectious disease with time fractional derivative and social behavior

The main purpose of this paper is to explore the influence of predation on the spread of a disease developed in the prey population where we assume that the prey has a social behavior.The memory of the prey and the predator measured by the time fractional derivative plays a crucial role in modeling the dynamical response in a predator–prey interaction.This memory can be modeled to articulate the involvement of interacting species by the presence of the time fractional derivative in the considered models.For the purpose of studying the complex dynamics generated by the presence of infection and the time-fractional-derivative we split our study into two cases. The first one is devotedto study the effect of a non-selective hunting on the spread of the disease, where the localstability of the equilibria is investigated. Further the backward bifurcation is obtainedconcerning basic reproduction rate of the infection. The second case is for explaining theimpact of selecting the weakest infected prey on the edge of the herd by a predator onthe prevalence of the infection, where the local behavior is scrutinized. Moreover, for thegraphical representation part, a numerical simulation scheme has been achieved usingthe Caputo fractional derivative operator.