A Complex Dynamic of an Eco‐Epidemiological Mathematical Model with Migration

In this paper, we present and analyze predator-prey system where prey population is linearly harvested and affected by fear and the prey population has grown logistically in the absence of predators. The predator population follows hunting cooperation, and it predates the prey population in the Holling type II functional responses. Based on those assumptions, a two-dimensional mathematical model is derived. The positivity, boundedness, and extinction of both prey and predator populations of the solution of the system are discussed. The existence, stability (local and global), and the Hopf bifurcation analysis of the biologically feasible equilibrium points are investigated. The aim of this research is to explore the effect of fear on the prey population and hunting cooperation on the predator population, and both prey and predator populations are harvested linearly and taken as control parameters of the model. If the values of c 1 > 1 , then both prey and predator populations are extinct and also fear parameter has a stabilizing effect on system 4. From the numerical simulation, it was found that the fear effect, hunting cooperation, prey harvesting, and predator harvesting change the dynamics of system 4.

Understanding predator-prey dynamics is essential for maintaining ecological balance and biodiversity. Classical models often fail to capture complex biological behaviors such as prey defense mechanisms and nonlinear predation effects, which are vital for accurately describing real ecosystems. In light of this, there is a growing need to incorporate behavioral and functional complexity into mathematical models to better understand species interactions and their long-term ecological outcomes. This study aims to develop and analyze a predator-prey model that integrates two key ecological features: a Holling type III functional response and the anti-predator behavior exhibited by prey. The model assumes a habitat with limited carrying capacity to reflect environmental constraints. We formulate a nonlinear system of differential equations representing the interaction between prey and predator populations. The model is examined analytically by identifying equilibrium points and analyzing their local stability using the Routh-Hurwitz criteria. A literature-based theoretical analysis is supplemented with numerical simulations to validate and illustrate population dynamics. The model exhibits three equilibrium points: a trivial solution (extinction), a predator-free equilibrium, and a non-trivial saddle point representing coexistence. The non-trivial equilibrium best reflects ecological reality, indicating stable coexistence where prey consumption is balanced by reproduction, and predator mortality aligns with energy intake. Numerical simulations show that prey populations initially grow rapidly, then decline as they reach carrying capacity, while predator populations grow after a time lag and eventually stabilize. The results are further supported by the eigenvalue analysis, confirming local asymptotic stability. The proposed model realistically captures predator-prey dynamics, demonstrating that the inclusion of anti-predator behavior and a Holling type III response significantly affects population trajectories and system stability. This framework provides a more ecologically valid approach for studying long-term species coexistence and highlights the importance of incorporating behavioral responses in ecological modeling.

The predator-prey model described is a population growth model of an eco-epidemiological system with prey protection and predator intraspecific traits. Predation interactions in predator species use response functions. The aim of this research is to examine the local stable balance point and look at the characteristics of species resulting from mathematical modeling interventions. Review of balance point analysis, numerical simulation and analysis of given trajectories. The research results show the shape of the model which is arranged with a composition of 5 balance points. There is one rational balance point to be explained, using the Routh-Hurwitz criterion, . The characteristic equation and associated eigenvalues in the mathematical model are the local asymptotically stable balance points. In the trajectory analysis, local stability is also shown by the model formed. There are differences for each population to reach its point of stability. The role of prey protection behavior is very effective in suppressing the spread of disease. Meanwhile, intraspecific predator interactions are able to balance the decreasing growth of prey populations. If we increase the intraspecific interaction coefficient, we can be sure that the growth of the prey population will both increase significantly. When the number of prey populations increases significantly, of course disease transmission and prey protection become determining factors, the continuation of the model in exosite interactions. In prey populations and susceptible prey to infection, growth does not require a long time compared to the growth of predator populations. The time required to achieve stable growth is rapid for the prey species. Although prey species' growth is more fluctuating compared to predator populations. Predatory species are more likely to be stable from the start of their growth. The significance of predatory growth is only at the beginning of growth, while after that it increases slowly and reaches an ideal equilibrium point. Each species has its own characteristics, so extensive studies are needed on more complex forms of response functions in further research.

In this paper, our aim is mathematical analysis and numerical simulation of a prey-predator model to describe the effect of predation between prey and predator with nonlinear functional response. First, we develop results concerning the boundedness, the existence and uniqueness of the solution. Furthermore, the Lyapunov principle and the Routh–Hurwitz criterion are applied to study respectively the local and global stability results. We also establish the Hopf-bifurcation to show the existence of a branch of nontrivial periodic solutions. Finally, numerical simulations have been accomplished to validate our analytical findings.

A Leslie-Gower ecoepidemic model with disease in the predators is constructed and analyzed. The total population is subdivided into three subclasses, namely, susceptible predator, infected predator, and prey population. The positivity, boundness of solutions, and the existence of the equilibria are studied, and the sufficient conditions of local asymptotic stability of the equilibria are obtained by the Routh-Hurwitz criterion. We analyze the global stability of the interior equilibria by using Lyapunov functions. It is observed that a Hopf bifurcation may occur around the interior equilibrium. At last, numeric simulations are performed in support of the feasibility of the main result.

Stochastic prey–predator model with additional food for predator

The fear response is an important anti-predator adaptation that can significantly reduce prey's reproduction by inducing many physiological and psychological changes in the prey. Recent studies in behavioral sciences reveal this fact. Other than terrestrial vertebrates, aquatic vertebrates also exhibit fear responses. Many mathematical studies have been done on the mass mortality of pelican birds in the Salton Sea in Southern California and New Mexico in recent years. Still, no one has investigated the scenario incorporating the fear effect. This work investigates how the mass mortality of pelican birds (predator) gets influenced by the fear response in tilapia fish (prey). For novelty, we investigate a modified fractional-order eco-epidemiological model by incorporating fear response in the prey population in the Caputo-fractional derivative sense. The fundamental mathematical requisites like existence, uniqueness, non-negativity and boundedness of the system's solutions are analyzed. Local and global asymptotic stability of the system at all the possible steady states are investigated. Routh-Hurwitz criterion is used to analyze the local stability of the endemic equilibrium. Fractional Lyapunov functions are constructed to determine the global asymptotic stability of the disease-free and endemic equilibrium. Finally, numerical simulations are conducted with the help of some biologically plausible parameter values to compare the theoretical findings. The order $\alpha$ of the fractional derivative is determined using Matignon's theorem, above which the system loses its stability via a Hopf bifurcation. It is observed that an increase in the fear coefficient above a threshold value destabilizes the system. The mortality rate of the infected prey population has a stabilization effect on the system dynamics that helps in the coexistence of all the populations. Moreover, it can be concluded that the fractional-order may help to control the coexistence of all the populations.

By means of mathematical models and an experimental system we argue that density—dependent genetic feedback has the potential to regulate the amount of energy taken by a predator (parasite and herbivore) population from the prey or host population. The experimental system was a simulation using the housefly, Musca domestica, with artificial reproduction and prey, thereby restricting predator population feeding and adjusting it to surplus energy of the prey or host population. The experimental predator population, controlled by genetic feedback, was adjusted to a mean density of about 53 animals and in turn was removing less than the total amount of energy available from the prey population. The control predator population, regulated only by competition, fluctuated wildly about a mean density of about 147 animals, and consumed about 78% of the energy available from its prey population. Fluctuations existed in the experimental predator population, but were significantly damped compared with those in the control. These results generally complement those of earlier mathematical models, and support the idea that prey evolution can enhance predator stability and lead to a balanced supply—demand economy between predator and prey. In nature, the genetic feedback mechanism must function interdependently with other population control mechanisms, but the attempt here is to isolate the mechanism and assess its effect.

A predator–prey model is proposed in this work where the prey population is infected by a disease. Here, healthy prey species show defence mechanism while they are attacked by the predator. Moreover as the infected prey are already physically weak, so, predator apply cooperative hunting strategy while consume infected prey to get more food. It helps the predator population to grow with a higher rate. But calculation gives that if they start to hunt the infected prey with a larger cooperative hunting rate, then ultimately predator population decrease with time. Boundedness and positivity of the system variables show that the proposed model system is well-posed. Routh–Hurwitz criterion provides the local stability conditions of the equilibrium points. Also, the system becomes permanent under certain parametric restrictions. The numerical results, verified using MATLAB, support the analytical findings. Numerical simulations give that the parameter denoting cooperative hunting rate can change the system dynamics and we can get oscillating behaviour by regulating this parameter. Moreover transcritical and saddle-node bifurcations occur by regulating the death rate of predator around the critical points. Occurrence of Bogdanov–Takens, generalized Hopf and Cusp bifurcations have also been observed here.

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

In this work we have studied a stochastic predator-prey model where the prey grows logistically in the absence of predator. All parameters but carrying capacity have been perturbed with telephone noise. The prey’s growth rate and the predator’s death rate have also been perturbed with white noises. Both of these noises have been proved extremely useful to model rapidly fluctuating phenomena Dimentberg (1988). The conditions under which extinction of predator and prey populations occur have been established. We also give sufficient conditions for positive recurrence and the existence of an ergodic stationary distribution of the positive solution, red which in stochastic predator-prey systems means that the predator and prey populations can be persistent, that is to say, the predator and prey populations can be sustain a quantity that is neither too much nor too little. In our analysis, it is found that the environmental noise plays an important role in extinction as well as coexistence of prey and predator populations. It is shown in numerical simulation that larger white noise intensity will lead to the extinction of the population, while telephone noise may delay or reduce the risk of species extinction.

In this work we have studied a stochastic predator-prey model where the prey grows logistically in the absence of predator. All parameters but carrying capacity have been perturbed with telephone noise. The prey’s growth rate and the predator’s death rate have also been perturbed with white noises. Both of these noises have been proved extremely useful to model rapidly fluctuating phenomena Dimentberg (1988). The conditions under which extinction of predator and prey populations occur have been established. We also give sufficient conditions for positive recurrence and the existence of an ergodic stationary distribution of the positive solution, red which in stochastic predator-prey systems means that the predator and prey populations can be persistent, that is to say, the predator and prey populations can be sustain a quantity that is neither too much nor too little. In our analysis, it is found that the environmental noise plays an important role in extinction as well as coexistence of prey and predator populations. It is shown in numerical simulation that larger white noise intensity will lead to the extinction of the population, while telephone noise may delay or reduce the risk of species extinction.

ABSTRACT: The dynamic relationship between prey populations and predator populations can be represent in the mathematical model. This research was developed from the mathematical model of predator-prey which was first introduced by Lotka-Volterra, namely by increasing the realism of the model through apply the logistic model to the prey and involving Holling type-III response function. The effects of drought on both populations and constant harvesting of prey are also included in the mathematical model. After the mathematical model is formed, a nondimensional model is then carried out to create a shape from the model that was built previously. This study aims to analyze the impact of drought on the prey-prey system with a type III Holling functional response and constant-yield prey harvesting. There is at least one equilibrium point and there are at most three equilibrium points in the model. Numerical simulations are carried out on the model to see the phase portrait. The simulation results show that if the rate of drought is greater than the intrinsic growth rate of prey then both populations will go towards extinction. However, on the other hand, if the intrinsic growth rate of the prey is greater than the rate of drought, then the dynamics relationship between the predator and prey populations depends on the pattern of constant-yield prey harvesting. Thus, a stable dynamic relationship occurs on the constant-yield prey harvesting at the interval 0<h<=K(r-a1)2

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

<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>