A Leslie-Gower Holling Type-II Predator-Prey Mathematical Model with Disease in Prey Population Incorporating a Prey Refuge

We formulate and analyze a predator-prey model followed by Leslie-Gower model in which the prey population is infected by disease. We assume that the disease can only spread over prey population. As a result prey population has been classified into two categories, namely susceptible prey, infected prey where as the predator population remains free from infection. To investigate the behaviour of prey population we incorporate prey refuge in this model. Since the prey refuge decreases the predation rate then it has an important effect in our predator-prey interaction model. We have discussed the existence of various equilibrium points and local stability analysis at those equilibrium points. We investigate the Hopf-bifurcation analysis about the interior equilibrium point by taking the rate of infection parameter and the prey refuge parameter as bifurcation parameters. The numerical analysis is carried out to support the analytical results and to discuss some interesting results that our model exhibits.

Dynamical behaviors of a constant prey refuge ratio-dependent prey-predator model with Allee and fear effects

In this paper,we consider a nonlinear ratio-dependent prey-predator model with constant prey refuge in the prey population.Both Allee and fear phenomena are incorporated explicitly in the growth rate of the prey population.The qualitative behaviors of the proposed model are investigated around the equilibrium points in detail.Hopf bifurcation including its direction and stability for the model is also studied.We observe that fear of predation risk can have both stabilizing and destabilizing effects and induces bubbling phenomenon in the system.It is also observed that for a fixed strength of fear,an increase in the Allee parameter makes the system unstable,whereas an increase in prey refuge drives the system toward stability.However,higher values of both the Allee and prey refuge parameters have negative impacts and the populations go to extinction.Further,we explore the variation of densities of the populations in different bi-parameter spaces,where the coexistence equilibrium point remains stable.Numerical simulations are carried out to explore the dynamical behaviors of the system with the help of MATLAB software.

Hopf bifurcation in a delayed prey-predator model with prey refuge involving fear effect

This work investigates a prey-predator model featuring a Holling-type II functional response,in which the fear effect of predation on the prey species,as well as prey refuge,are considered.Specifically,the model assumes that the growth rate of the prey population decreases as a result of the fear of predators.Moreover,the detection of the predator by the prey species is subject to a delay known as the fear response delay,which is incorporated into the model.The paper establishes the preliminary conditions for the solution of the delayed model,including positivity,boundedness and permanence.The paper discusses the existence and stability of equilibrium points in the model.In particular,the paper considers the discrete delay as a bifurcation parameter,demonstrating that the system undergoes Hopf bifurcation at a critical value of the delay parameter.The direction and stability of periodic solutions are determined using central manifold and normal form theory.Additionally,the global stability of the model is established at axial and positive equilibrium points.An extensive numerical simulation is presented to validate the analytical findings,including the continuation of the equilibrium branch for positive equilibrium points.

Switching effect on a two prey-one predator system with strong Allee effect incorporating prey refuge

In an environment,the food chains are balanced by the prey-predator interactions.When a predator species is provided with more than one prey population,it avails the option of prey switching between prey species according to their availability.So,prey switching of predators mainly helps to increase the overall growth rate of a predator species.In this work,we have proposed a two prey-one predator system where the predator population adopts switching behavior between two prey species at the time of consumption.Both the prey population exhibit a strong Allee effect and the predator population is considered to be a generalist one.The proposed system is biologically well-defined as the system variables are positive and do not increase abruptly with time.The local stability analysis reveals that all the predator-free equilibria are saddle points whereas the prey-free equilibrium is always stable.The intrinsic growth rates of prey,the strong Allee parameters,and the prey refuge parameters are chosen to be the controlling parameters here.The numerical simulation reveals that in absence of one prey,the other prey refuge parameter can change the system dynamics by forming a stable or unstable limit cycle.Moreover,a situation of bi-stability,tri-stability,or even multi-stability of equilibrium points occurs in this system.As in presence of the switching effect,the predator chooses prey according to their abundance,so,increasing refuge in one prey population decreases the count of the second prey population.It is also observed that the count of predator population reaches a comparatively higher value even if they get one prey population at its fullest quantity and only a portion of other prey species.So,in the scarcity of one prey species,switching to the other prey is beneficial for the growth of the predator population.

Dynamical Behaviors of a Modified Leslie-Gower Predator-Prey System with Fear Effect and Prey Refuge

In this paper, the dynamical behaviors of a modified Leslie-Gower predator-prey model incorporating fear effect and prey refuge are investigated. We delve into the construction of the model and its biological significance, with preliminary results encompassing positivity, boundedness, and persistence. The stability of the system’s boundary and positive equilibrium points is proven by calculating the real part of the eigenvalues of the Jacobian matrix. At the positive equilibrium point, we demonstrate that the system’s unique positive equilibrium is globally asymptotically stable by using the Dulac criterion. Furthermore, at this equilibrium point, we employ the Implicit Function Theorem to discuss how fear effects and prey refuges influence the population densities of both prey and predators. Finally, numerical simulations are conducted to validate the above-mentioned conclusions and explored the impact of Predator-taxis sensitivity αon dynamics of the system.

Existence of spatiotemporal patterns in the reaction-diffusion predator-prey model incorporating prey refuge

The pattern formation in reaction-diffusion system has long been the subject of interest to the researchers in the domain of mathematical ecology because of its universal exis- tence and importance. The present investigation deals with a spatial dynamics of the Beddington-DeAngelis predator-prey model in the presence of a constant proportion of prey refuge. The model system representing boundary value problem under study is subjected to homogeneous Neumann boundary conditions. The asymptotic stability including the local and the global stability and the bifurcation as well of the unique pos- itive homogeneous steady state of the corresponding temporal model has been analyzed. The Turing instability region in two-parameter space and the condition of diffusion- driven instability of the spatiotemporal model are investigated. Based on the appro- priate numerical simulations, the present model dynamics in Turing space appears to get influenced by prey refuge while it exhibits diffusion-controlled pattern formation growth to spots, stripe-spot mixtures, labyrinthine, stripe-hole mixtures and holes repli- cation. The results obtained appear to enrich the findings of the model system under consideration.

NONCONSTANT PREY HARVESTING IN RATIO-DEPENDENT PREDATOR-PREY SYSTEM INCORPORATING A CONSTANT PREY REFUGE

This paper deals with the problem of nonconstant harvesting of prey in a ratio-dependent predator-prey system incorporating a constant prey refuge. Here we use the reasonable catch-rate function instead of usual catch-per-unit-effort hypothesis. The existence, as well as the stability of possible equilibria, is carried out. Bionomic equilibrium of the system is determined and optimal harvest policy is studied with the help of Pontryagin＇s maximum principle. The key results developed in this paper are illustrated using numer- ical simulations. Our results indicate that dynamic behavior of the system very much depends on the prey refuge parameter and increasing amount of refuge could increase prey density and may lead to the extinction of predator population density.

Bifurcation and Limit Cycle of a Ratio-dependent Predator-prey, System with Refuge on Prey

Influences of prey refuge on the dynamics of a predator-prey model with ratio-dependent functional response are investigated. The local and global stability of positive equilibrium of the system are considered. Theoretical analysis indicates that constant refuge leads to the system undergo supercritical Hopf bifurcation twice with the birth rate of prey species changing continuously.

GLOBAL DYNAMICS OF A PREDATOR-PREY MODEL WITH PREY REFUGE AND DISEASE

In this paper, we study a predator-prey model with prey refuge and disease. We study the local asymptotic stability of the equilibriums of the system. Further, we show that the equilibria are globally asymptotically stable if the equilibria are locaUy asymptotically stable. Some examples are presented to verify our main results. Finally, we give a brief discussion.

HOPF BIFURCATION OF AN INFECTED PREDATOR-PREY SYSTEM WITH PREY REFUGE

In this paper, an infected predator-prey model with prey refuge is investigated. The effects of refuge on the stability of the equilibria of the system are analyzed. Moreover, using the criterion introduced by Liu, we derive the Hopf bifurcation conditions of the system with respect to the refuge value.

Prey-predator model with functional response for optimal harvesting incorporating prey refuge

This paper presents a prey-predator model considering the predator interacting with non-refuges prey by class of functional responses. Here we also consider harvesting for only non-refuges prey. We discuss the equilibria of the model, and their stability for hiding prey either in constant form or proportional to the densities of prey population. We also investigate various possibilities of bionomic equilibrium and optimal harvesting policy. Finally we present numerical examples with pictorial presentation of the various effects of the prey predator system parameter.

THE EFFECT OF REFUGE AND PROPORTIONAL HARVESTING FOR A PREDATOR-PREY SYSTEM WITH REACTION-DIFFUSION

A diffusive predator-prey system with Holling-Tanner functional response and no-flux boundary condition is considered in this work.By using upper and lower solutions combined with iteration method,sufficient condition which ensures the global asymptotical stability of the unique positive equilibrium of the system is obtained.It is shown that the prey refuge and the proportional harvesting can influence the global asymptotical stability of unique positive equilibrium of the system,furthermore,they can change the position of the unique interior equilibrium and make species coexist more easily.