DETAILED ANALYSIS OF AN ECO-EPIDEMIOLOGICAL SYSTEM WITH PREY REFUGE, FEAR EFFECT AND COMPETITION AMONG THE PREDATOR SPECIES

<p>In this article, the dynamical behavior of a three dimensional continuous time eco-epidemiological model is studied. A prey-predator model involving infectious disease in predator population is proposed and analyzed. This model deals with SI infectious disease that transmitted horizontally in predator population. It is assumed that the disease transmitted to susceptible population in two different ways: contact with infected individuals and an external sources. The existence, uniqueness and bounded-ness of the solution of this model are investigated. The local and global stability conditions of all possible equilibrium points are established. The local bifurcation analysis and a Hopf bifurcation around the positive equilibrium point are obtained. Finally, numerical simulations are given to illustrate our obtained analytical results.</p>

In this study, the different dynamical behaviors caused by different parameters of a discrete-time eco-epidemiological model with disease in prey are discussed in ecological perspective. The results indicate that when we choose the same parameters and initial value and only vary the key parameters there appears a series of dynamical behaviors. For example, only varying the death rate of the infected prey (the carrying capacity of the environment for the prey population or the transmission coefficient), there appear chaos, Hopf (flip) bifurcation, local stability, flip (Hopf) bifurcation, and chaos; when only varying the predation coefficient there appear chaos, Hopf bifurcation, local stability, Hopf bifurcation, and chaos. These results are far richer than the corresponding continuous-time model and are rarely seen in previous works. Numerical simulations not only illustrate our results but also exhibit complex dynamical behaviors, such as period-doubling bifurcation in period-2,4,8, quasi-periodic orbits, 3,5,11,16-period orbits and chaotic sets. Moreover, the numerical simulations imply that when the death rate of the infected prey reaches a fixed value the disease dies out. Also, when the predation coefficient parameter reaches some value the disease dies out. These findings indicate that it is practicable to control the disease transmitting in prey by changing the death rate of the infected prey and the predation coefficient parameter.

<p style='text-indent:20px;'>In this paper, a predator-prey model with age structure and ratio-dependent response function incorporating a prey refuge is investigated. The model is formulated as an abstract non-densely defined Cauchy problem and a sufficient condition for the existence of the positive age-related equilibrium is given. Then using the integral semigroup theory and the Hopf bifurcation theory for semilinear equations with non-dense domain, it is shown that Hopf bifurcation occurs at the positive age-related equilibrium. Numerical simulations are performed to validate theoretical results and sensitivity analyses are presented. The results show that the prey refuge has a stabilizing effect, that is, the prey refuge is an important factor to maintain the balance between prey and predator population.</p>

In this work, we propose an eco-epidemiological predator–prey interactions model in a fuzzy environment. The model incorporates the influence of environmental uncertainty on the interaction between predator and prey populations, along with the effects of disease on the predator population. We use a combination of fuzzy set theory and differential equations to model the interactions between the predator and prey populations. Subsequently, our analysis demonstrates that the uncertain environmental conditions can disrupt the system’s stability by modifying critical values of model parameters. Furthermore, we have investigated the Hopf bifurcation by using the rate of infection as a bifurcation parameter. Overall, our study highlights the importance of considering both environmental fuzziness and disease in predator–prey models and provides insights into the complex dynamics that can arise in such systems. We also perform numerical simulations to give more insights for our analytical findings.

We study a predator-prey model with different characteristic time scales for the prey and predator populations, assuming that the predator dynamics is much slower than the prey one. Geometrical Singular Perturbation theory provides the mathematical framework for analyzing the dynamical properties of the model. This model exhibits a Hopf bifurcation and we prove that when this bifurcation occurs, a canard phenomenon arises. We provide an analytic expression to get an approximation of the bifurcation parameter value for which a maximal canard solution occurs. The model is the well-known Rosenzweig-MacArthur predator-prey differential system. An invariant manifold with a stable and an unstable branches occurs and a geometrical approach is used to explicitly determine a solution at the intersection of these branches. The method used to perform this analysis is based on Blow-up techniques. The analysis of the vector field on the blown-up object at an equilibrium point where a Hopf bifurcation occurs with zero perturbation parameter representing the time scales ratio, allows to prove the result. Numerical simulations illustrate the result and allow to see the canard explosion phenomenon.

Hopf bifurcation of an age-structured prey–predator model with Holling type II functional response incorporating a prey refuge

In this paper, we consider a two-predator–one-prey population model that incorporates both the inter-specific competition between two predator populations and the intra-specific competition within each predator population. We investigate the dynamics of this model by addressing the existence, local and global stability of equilibria, uniform persistence as well as saddle-node and Hopf bifurcations. Numerical simulations are presented to explore the joint impacts of inter-specific and intra-specific competition on competition outcomes. Though inter-specific competition along does not admit a stable coexistence equilibrium, with intra-specific competition, the coexistence of the two competing predator species becomes possible and the two coexisting predator species may maintain at two different equilibrium populations.

<p style='text-indent:20px;'>In this paper, we study the dynamics of a Leslie-Gower predator-prey system with hunting cooperation among predator population and constant-rate harvesting for prey population. It is shown that there are a weak focus of multiplicity up to three and a cusp of codimension at most two for various parameter values, and the system exhibits two saddle-node bifurcations, a Bogdanov-Takens bifurcation of codimension two and a Hopf bifurcation as the bifurcation parameters vary. The results developed in this article reveal far more complex dynamics compared to the Leslie-Gower system and show how the prey harvesting and the hunting cooperation affect the dynamics of the system. In particular, there exist some critical values of prey harvesting and hunting cooperation such that the predator and prey populations are at risk of extinction if the intensities of harvesting and hunting cooperation are greater than these critical values. Moreover, numerical simulations are presented to illustrate our theoretical results.</p>

<p>In this paper, we proposed and studied a Leslie-Gower prey-predator system which considered various ecological factors, such as the Allee effect and harvesting effect in prey populations and the hunting cooperation in predator populations. The positivity and boundedness of the system's solutions were determined. The conditions for the uniformly persistence of the system and the extinction of populations have been established. We studied the existence and type of singularities, including primary singularities and higher-order singularities. Using topological equivalent and the blow-up method, we proved that the origin was the attractor, and a defined basin of attraction was given. As the parameters change, the system may experience two saddle-node bifurcations and a Hopf bifurcation. The direction and stability of Hopf bifurcation solutions were established. Numerical simulations were given to validate the primary theoretical findings. In this paper, we found that there existed critical thresholds for Allee threshold, prey harvesting, and hunting cooperation, beyond which both predator and prey populations faced the risk of extinction.</p>

<abstract><p>We consider a delayed diffusive predator-prey system with nonlocal competition in prey and schooling behavior in predator. We mainly study the local stability and Hopf bifurcation at the positive equilibrium by using time delay as the parameter. We also analyze the property of Hopf bifurcation by center manifold theorem and normal form method. Through the numerical simulation, we obtain that time delay can affect the stability of the positive equilibrium and induce spatial inhomogeneous periodic oscillations of prey and predator's population densities. In addition, we observe that the increase of space area will not be conducive to the stability of the positive equilibrium $ (u_*, v_*) $, and may induce the inhomogeneous periodic oscillations of prey and predator's population densities under some values of the parameters.</p></abstract>

In this paper, we study the spatiotemporal dynamics of a diffusive Leslie-type predator–prey system with Beddington–DeAngelis functional response under homogeneous Neumann boundary conditions. Preliminary analysis on the local asymptotic stability and Hopf bifurcation of the spatially homogeneous model based on ordinary differential equations is presented. For the diffusive model, firstly, it is shown that Turing (diffusion-driven) instability occurs which induces spatial inhomogeneous patterns. Next, it is proved that the diffusive model exhibits Hopf bifurcation which produces temporal inhomogeneous patterns. Furthermore, at the points where the Turing instability curve and Hopf bifurcation curve intersect, it is demonstrated that the diffusive model undergoes Turing–Hopf bifurcation and exhibits spatiotemporal patterns. Numerical simulations are also presented to verify the theoretical results.

Effects of incubation and gestation periods in a prey–predator model with infection in prey

Spatiotemporal dynamics of Leslie–Gower predator–prey model with Allee effect on both populations

Global boundedness and dynamics of a diffusive predator–prey model with modified Leslie–Gower functional response and density-dependent motion

The harvesting of species occurs in terrestrial and aquatic habitats across the world. It not only causes alteration in the population structure of the species subjected to harvesting but also of the species in interaction with the harvested species. The present work investigates the effect of nonlinear prey harvesting on the dynamics of a ratio-dependent predator–prey system with a strong Allee effect in prey population. It is found that the system exhibits a rich spectrum of dynamics including saddle-node bifurcation, Hopf bifurcation and homoclinic bifurcation with respect to the parameters that shape the nonlinear harvesting rate, namely, the maximum harvesting rate and a half-saturation constant that represents the prey density at which half of the maximum harvesting rate is reached. It is found that the basin of attraction of the stable coexistence state shrinks as the harvesting rate increases and if the harvesting rate is above a threshold value at which saddle-node bifurcation occurs, the stable coexistence of predator and prey populations is not possible for any initial start. It is also found that the harvesting policies in which the harvesting rate increases less rapidly at low prey population size are more favorable for the stable coexistence of species. The presence of Allee effect in the prey population is found to increase the chances of extinction of both species by reducing the threshold value of the harvesting rate at which the unconditional extinction occurs. Numerical simulations are carried out to support the analytical findings.