Wave of chaos in a spatial eco-epidemiological system: Generating realistic patterns of patchiness in rabbit–lynx dynamics (original) (raw)
Related papers
Spread of a disease and its effect on population dynamics in an eco-epidemiological system
Communications in Nonlinear Science and Numerical Simulation, 2014
In this paper, an eco-epidemiological model with simple law of mass action and modified Holling type II functional response has been proposed and analyzed to understand how a disease may spread among natural populations. The proposed model is a modification of the model presented by Upadhyay et al. [1]. Existence of the equilibria and their stability analysis (linear and nonlinear) has been studied. The dynamical transitions in the model have been studied by identifying the existence of backward Hopf-bifurcations and demonstrated the period-doubling route to chaos when the death rate of predator (1 µ) and the growth rate of susceptible prey population (r) are treated as bifurcation parameters. Our studies show that the system exhibits deterministic chaos when some control parameters attain their critical values. Chaotic dynamics is depicted using the 2D parameter scans and bifurcation analysis. Possible implications of the results for disease eradication or its control are discussed.
Evidence of chaos in eco-epidemic models
Mathematical Biosciences and Engineering, 2009
We study an eco-epidemic model with two trophic levels in which the dynamics is determined by predator-prey interactions as well as the vulnerability of the predator to a disease. Using the concept of generalized models we show that for certain classes of eco-epidemic models quasiperiodic and chaotic dynamics is generic and likely to occur. This result is based on the existence of bifurcations of higher codimension such as double Hopf bifurcations. We illustrate the emergence of chaotic behavior with one example system.
Dynamical bifurcation of an eco-epidemiological system
In this paper we have considered a dynamical system which models the evolution of three species, two of them being the susceptible and infected preys, respectively and the other one-the predator population. The system depends on eight parameters. We found a Hopf bi-furcation point when one of the parameters was varied and we deduced the presence of a stable limit cycle. Bifurcation diagrams are presented and we established the important types of dynamics of the system. By numerical integration, we obtained the phase portrait for different types of dynamics and plots of time course for the corresponding solutions. M.S.C. 2010: 37C75, 34K18.
Deterministic Chaos vs. Stochastic Fluctuation in an Eco-epidemic Model
Mathematical Modelling of Natural Phenomena, 2012
An eco-epidemiological model of susceptible Tilapia fish, infected Tilapia fish and Pelicans is investigated by several author based upon the work initiated by Chattopadhyay and Bairagi (Ecol. Model., 136, 103-112, 2001). In this paper, we investigate the dynamics of the same model by considering different parameters involved with the model as bifurcation parameters in details. Considering the intrinsic growth rate of susceptible Tilapia fish as bifurcation parameter, we demonstrate the period doubling route to chaos. Next we consider the force of infection as bifurcation parameter and demonstrate the occurrence of two successive Hopfbifurcations. We identify the existence of backward Hopf-bifurcation when the death rate of predators is considered as bifurcation parameter. Finally we construct a stochastic differential equation model corresponding to the deterministic model to understand the role of demographic stochasticity. Exhaustive numerical simulation of the stochastic model reveals the large amplitude fluctuation in the population of fish and Pelicans for certain parameter values. Extinction scenario for Pelicans is also captured from the stochastic model.
The effect of parasites and pathogens in the prey population received a considerable amount of population has been paid little attention. In this study, we consider a predator-prey model with disease in the predator population and the disease transmission is assumed to be asymptotic in nature. To observe the dynamics of the system the local stability analysis around the biologically feasible equilibria is performed. We also derive the ecological as well as the disease basic reproduction numbers and analyze the community structure of the model system in terms of these numbers. Our numerical results reveal that disease introduction in the predator population produces chaotic dynamics. We observe disease free equilibrium, limit cycles, period-doubling and chaos for variation of the force of infection in the predator population. We also observe that half-saturation constants are responsible for the occurrence and control of chaos. It is found that chaos may be prevented by increasing th...
Role of horizontal incidence in the occurrence and control of chaos in an eco-epidemiological system
Mathematical Medicine and Biology-a Journal of The Ima, 2007
A predator-prey model with disease in the prey population is proposed and analysed. The mode of disease transmission plays an important role in such dynamics. Keeping this factor in mind, we observe the dynamics of such a system for simple mass action incidence and standard incidence. Our observations indicate that the phenomenon of rarity or non-occurrence of chaos in our proposed model is well defined if the mode of disease transmission follows standard incidence. Moreover, using the method of Latin hypercube sampling, we show that the region of stability increases if the disease transmission follows the standard incidence law.
The spatial patterns through diffusion-driven instability in a predator–prey model
Applied Mathematical Modelling, 2012
Studies on stability mechanism and bifurcation analysis of a system of interacting populations by the combined effect of self and cross-diffusion become an important issue in ecology. In the current investigation, we derive the conditions for existence and stability properties of a predator-prey model under the influence of self and cross-diffusion. Numerical simulations have been carried out in order to show the significant role of self and cross-diffusion coefficients and other important parameters of the system. Various contour pictures of spatial patterns through Turing instability are portrayed and analysed in order to substantiate the applicability of the present model. Finally, the paper ends with an extended discussion of biological implications of our findings.
DYNAMICAL ANALYSIS OF AN ECO-EPIDEMIOLOGICAL MODEL EXPERIENCING THE CROWDING EFFECT OF INFECTED PREY
2024
Most eco-epidemiological models use a bi-linear functional response, also known as the simple law of mass action, to describe the transmission of an infection. The non-linear incidence rate considers the infected individuals' crowding effect and prevents the contact rate's unboundedness by choosing suitable parameters. This paper aims to construct an Eco-Epidemiological model following the nonlinear incidence rate suggested by Gumel and Moghadas 2003. The model also offers a reasonable, realistic approach to the ecological systems in the world as we follow the Holling type II for the predator-susceptible prey interaction and the simple mass action low for the predator for the predator-infected prey interaction as the infected prey would be weak. The time for finding it would be significantly more than the time needed to catch the healthy prey. We proved the solutions' positivity and existence and our model's boundedness. The equilibrium points are determined with the feasibility conditions for each. Local stability has been analysed using Routh Hurwitz, and a Lyapunov function has been constructed to study global stability according to La Salle theorem. Different types of bifurcation are observed using Sotomayor's and Hopf theorems. The numerical analysis of the solution was carried out using fourth-order Runge-Kutta. The simulations that we performed using MATLAB 2022a supported our theoretical findings.
Backward bifurcation, oscillations and chaos in an eco-epidemiological model with fear effect
Journal of Biological Dynamics
This paper considers an eco-epidemiological model with disease in the prey population. The disease in the prey divides the total prey population into two subclasses, susceptible prey and infected prey. The model also incorporates fear of predator that reduces the growth rate of the prey population. Furthermore, fear of predator lowers the activity of the prey population, which reduces the disease transmission. The model is well-posed with bounded solutions. It has an extinction equilibrium, susceptible prey equilibrium, susceptible prey-predator equilibrium, and coexistence equilibria. Conditions for local stability of equilibria are established. The model exhibits fear-induced backward bifurcation and bistability. Extensive numerical simulations show the presence of oscillations and occurrence of chaos due to fear induced lower disease transmission in the prey population.
Journal of Biological Systems, 2010
Eco-epidemiological models are now receiving much attention to the researchers. In the present article we re-visit the model of Holling-Tanner which is recently modified by Haque and Venturino 1 with the introduction of disease in prey population. Density dependent disease-induced predator mortality function is an important consideration of such systems. We extend the model of Haque and Venturino 1 with density dependent disease-induced predator mortality function. The existence and local stability of the equilibrium points and the conditions for the permanence and impermanence of the system are worked out. The system shows different dynamical behaviour including chaos for different values of the rate of infection. The model considered by Haque and Venturino 1 also exhibits chaotic nature but they did not shed any light in this direction. Our analysis reveals that by controlling disease-induced mortality of predator due to ingested infected prey may prevent the occurrence of chaos.