A New Deterministic Model for the Interaction between Predator and Prey

Throughout the history of ecology, the interaction between predators and their prey has received attention from ecologists. Some of the longest and most well-known data series in ecology are from predator and prey popula-tions, and predator–prey models are among the oldest in the field. Despite

Effects of deterministic and random refuge in a prey–predator model with parasite infection

Predator-prey models are essential for understanding ecological dynamics, and fractional-order models provide a more realistic approach by considering memory effects. This study aims to analyze the discrete dynamics of a predator-prey model, incorporating predator cannibalism, refuge, and memory effects with a Caputo-type fractional-order. The Piecewise Constant Argument (PWCA) method was employed for discretization, followed by an analysis of the equilibrium points and their stability. Four equilibrium points were identified: the origin, prey extinction, predator extinction, and coexistence. It was found that the origin point was unstable, while the prey extinction, predator extinction, and coexistence points were conditionally locally asymptotically stable, depending on the parameter values. The order of the fractional derivative and step size significantly influenced the stability of these equilibrium points. Numerical simulations confirmed the theoretical findings, showing how parameter variations affect system behavior.

This study discusses the intervention of cannibalism and disease spread with Holling Type II response function in the predator-prey model. It is assumed that disease infection is limited to the prey population and cannot be cured so that in this model there are three subpopulations namely susceptible prey, infected prey and predators. In addition, there is cannibalism in the predator population. The objectives of this study include constructing a predator-prey model with cannibalism intervention and disease infection in prey using Holling Type II response function, identifying the stability of the equilibrium point of the model and interpreting the model based on simulation results. Analysis of the stability of the equilibrium point is carried out with a linearization approach and the Routh-Hurwitz criterion was used to determine equilibrium stability. Based on the stability analysis, 5 (five) equilibrium points are obtained, namely population extinction, susceptible prey exists, predator extinction, infected prey extinction and population exists where the population extinction equilibrium point is unstable and the other equilibrium points are stable with the certain conditions. From the simulation, it is obtained that the numerical results are in accordance with the analytical results of the stability analysis of the equilibrium point of the model and for infinite time, there will be no population extinction while the state of susceptible prey exists, predator extinction, infected prey extinction and population exists can occur if the stability conditions are met. Based on the numerical simulations, it was found that changes in the parameter values of the rate of change of susceptible prey to infected prey and the coefficient of predator cannibalism in day-1 can cause changes in the type of stability of the equilibrium point. Thus, rate of change susceptible prey to infected prey and the coefficient of predator cannibalism affects the population of prey and predator.

A non-smooth switched harvest on predators is introduced into a simple predator-prey model with logistical growth of the prey and a bilinear functional response. If the density of the predator is below a switched value, the harvesting rate is linear; otherwise, it is constant. The model links the well studied predator-prey model with constant harvesting to that with a proportional harvesting rate. It is shown that when the net reproductive number for the predator is greater than unity, the system is permanent and there may exist multiple positive equilibria due to the effects of the switched harvest, a saddle-node bifurcation, a limit cycle, and the coexistence of a stable equilibrium and a unstable circled inside limit cycle and a stable circled outside limit cycle. When the net reproductive number is less than unity, a backward bifurcation from a positive equilibrium occurs, which implies that the stable predator-extinct equilibrium may coexist with two coexistence equilibria. In this situation, reducing the net reproductive number to less than unity is not enough to enable the predator to go extinct. Numerical simulations are provided to illustrate the theoretical results. It seems that the model possesses new complex dynamics compared to the existing harvesting models.

Macroscopic Dynamic Effects of Migrations in Patchy Predator-prey Systems

A dynamic systems approach can predict steady states in predator-prey interactions, but there are very few examples of predictions from predator-prey models conforming to empirical data. Here, we examined the evidence for the low-density steady state predicted by a Lotka-Volterra model of a crab-clam predator-prey system using data from long-term monitoring, and data from a previously published field survey and field predation experiment. Changepoint analysis of time series data indicate that a shift to low density occurred for the soft-shell clam Mya arenaria in 1972, the year of Tropical Storm Agnes. A possible mechanism for the shift is that Agnes altered predator-prey dynamics between M. arenaria and the blue crab Callinectes sapidus, shifting from a system controlled from the bottom up by prey resources, to a system controlled from the top down by predation pressure on bivalves, which is supported by a correlation analysis of time series data. Predator-prey ordinary differential equation models with these 2 species were analyzed for steady states, and low-density steady states were similar to previously published clam densities and mortality rates, consistent with the idea that C. sapidus is a major driver of M. arenaria population dynamics. Relatively simple models can predict shifts to alternative stable states, as shown by agreement between model predictions (this study) and published field data in this system. The preponderance of multispecies interactions exhibiting nonlinear dynamics indicates that this may be a general phenomenon.

Fitting a predator–prey model to zooplankton time-series data in the Gironde estuary (France): Ecological significance of the parameters

Impact of Allee effect on an eco-epidemiological system

Field theory tools are applied to analytically study fluctuation and correlation effects in spatially extended stochastic predator–prey systems. In the mean-field rate equation approximation, the classic Lotka–Volterra model is characterized by neutral cycles in phase space, describing undamped oscillations for both predator and prey populations. In contrast, Monte Carlo simulations for stochastic two-species predator–prey reaction systems on regular lattices display complex spatio-temporal structures associated with persistent erratic population oscillations. The Doi–Peliti path integral representation of the master equation for stochastic particle interaction models is utilized to arrive at a field theory action for spatial Lotka–Volterra models in the continuum limit. In the species coexistence phase, a perturbation expansion with respect to the nonlinear predation rate is employed to demonstrate that spatial degrees of freedom and stochastic noise induce instabilities toward structure formation, and to compute the fluctuation corrections for the oscillation frequency and diffusion coefficient. The drastic downward renormalization of the frequency and the enhanced diffusivity are in excellent qualitative agreement with Monte Carlo simulation data.

Classical predator-prey models usually emphasize direct predation as the primary means of interaction between predators and prey. However, several field studies and experiments suggest that the mere presence of predators nearby can reduce prey density by forcing them to adopt costly defensive strategies. Adoption of such kind would cause a substantial change in prey demography. The present paper investigates a predator-prey model in which the predator's consumption rate (described by a functional response) is affected by both prey and predator densities. Perceived fear of predators leads to a drop in prey's birth rate. We also consider both constant and time-varying (seasonal) forms of prey's birth rate and investigate the model system's respective autonomous and nonautonomous implementations. Our analytical studies include finding conditions for the local stability of equilibrium points, the existence, direction of Hopf bifurcation, etc. Numerical illustrations include bifurcation diagrams assisted by phase portraits, construction of isospike and Lyapunov exponent diagrams in bi-parametric space that reveal the rich and complex dynamics embedded in the system. We observe different organized periodic structures within the chaotic regime, multistability between multiple pairs of coexisting attractors with intriguing basins of attractions. Our results show that even relatively slight changes in system parameters, perturbations, or environmental fluctuations may have drastic consequences on population oscillations. Our observations indicate that the fear effect alters the system dynamics significantly and drives an otherwise irregular system toward regularity.

The seemingly paradoxical increase of a species population size in response to an increase in its mortality rate has been observed in several continuous-time and discrete-time models. This phenomenon has been termed the "hydra effect". In light of the fact that there is almost no empirical evidence yet for hydra effects in natural and laboratory populations, we address the question whether the examples that have been put forward are exceptions, or whether hydra effects are in fact a common feature of a wide range of models. We first propose a rigorous definition of the hydra effect in population models. Our results show that hydra effects typically occur in the well-known Gause-type models whenever the system dynamics are cyclic. We discuss the apparent discrepancy between the lack of hydra effects in natural populations and their occurrence in this standard class of predator-prey models.

On Volterra’s Population Equation

Social scientists have attempted in vain to explain and predict the social phenomenon and particularly the behavior of the social system, with the unsatisfactory result that they were not so successful in terms of the accuracy of the prediction that they started to look into chaos theory. Several authors presented the ability of even the most simple predator-prey models to yield damped and explosive oscillations as well as stable limit cycles. Lotka and Volterra suggested models of population dynamics incorporating interpopulation competition. In this paper, biological population ecology model, especially Lotka-Volterra model is applied to organizations and social systems at large. This paper demonstrates the power of merging system dynamics with population ecology models to assess the sensitivity to initial conditions. The dynamical properties of the generalized Lotka-Volterra model were made by simulations using the Ithink software. The implications of using simulation in the analysis of chaotic behavior are presented.

A predator-prey model with satiation and intraspecific competition