Predator-prey Population Research Articles

Modeling and Stability Analysis of the Dynamics of Prey-Predator Population

The aim of this work is to investigate the numerical study of prey-predator model and give a thorough overview of the existing research. Here we see instances that involve stability analysis, phase plane, and phase portrait, among others. Furthermore, a numerical simulation is shown to illustrate the results of the prey-predator model. This is to provide light on the procedures followed in order to arrive at the conclusions. The one-prey, one-predator principle is discussed in this article from multiple theoretical perspectives. Finally, for various parameter values and time intervals, we have demonstrated the phase portraits and the dynamic behavior of the pre-predator model. Jagannath University Journal of Science, Volume 11, Number 1, June 2024, pp. 81−93

Stationary distribution and mean extinction time in a generalist prey–predator model driven by Lévy noises

In this study, a delayed generalist prey–predator model incorporating the logistic-like growth and Holling type-III functional response for predator subjected to two Lévy noises is established. Such environmental fluctuations can describe abrupt change in population density. First, we conduct stability analysis and determine the critical value of time delay when Hopf bifurcation takes place. Then, the existence of the global positive solution and stability in distribution of stochastic model are studied. Numerical results are given to illustrate the analytical findings, and mainly dedicated to the consequences of time delay and Lévy noises on the stationary distribution, mean extinction time and reproduction rate of prey–predator populations, via stationary probability distribution function (PDF), mean first passage time (MFPT) and relaxation time, respectively. Our findings indicate that the decreases in time delay, noise intensities (Di,i=1,2), and stability indexes within the range of αi∈[1,2] are more likely to maintain the population in a high density state. Phase transition occurs when the system is in the suitable parameter regime. Additionally, there exist optimal noise intensities that slow down switching of prey species from a stable state to an extinction state. Smaller time delay τ and stability indexes αi inhibit population extinction, whereas smaller skewness parameter β speed up prey extinction. More importantly, two noises can tune the enhanced stability characteristic of the system.

Asymptotic spreading of predator-prey populations in a shifting environment

Inspired by a recent study associating shifting temperature conditions with changes in the efficiency with which predators convert prey to offspring, we propose a predator prey model of reaction-diffusion type to analyze the consequence of such effects on the population dynamics and spread of {the predator} species. In the model, the predator conversion efficiency is represented by a spatially heterogeneous function depending on the variable $\xi=x-c_1t$ for some given $c_1>0$. Using the Hamilton-Jacobi approach, we provide explicit formulas for the spreading speed of the predator species. When the conversion function is monotone increasing, the spreading speed is determined in all cases and non-local pulling is possible. When the function is monotone decreasing, we provide formulas for the spreading speed when the rate of shift of the conversion function is sufficiently fast or slow.

Predator–Prey Interaction with Fear Effects: Stability, Bifurcation and Two-Parameter Analysis Incorporating Complex and Fractal Behavior

This study investigates the dynamics of predator–prey interactions with non-overlapping generations under the influence of fear effects, a crucial factor in ecological research. We propose a novel discrete-time model that addresses limitations of previous models by explicitly incorporating fear. Our primary question is: How does fear influence the stability of predator–prey populations and the potential for chaotic dynamics? We analyze the model to identify biologically relevant equilibria (fixed points) and determine the conditions for their stability. Bifurcation analysis reveals how changes in fear levels and predation rates can lead to population crashes (transcritical bifurcation) and complex population fluctuations (period-doubling and Neimark–Sacker bifurcations). Furthermore, we explore the potential for controlling chaotic behavior using established methods. Finally, two-parameter analysis employing Lyapunov exponents, spectrum, and Kaplan–Yorke dimension quantifies the chaotic dynamics of the proposed system across a range of fear and predation levels. Numerical simulations support the theoretical findings. This study offers valuable insights into the impact of fear on predator–prey dynamics and paves the way for further exploration of chaos control in ecological models.

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

In this paper, we propose an eco-epidemiological mathematical model in order to describe the effect of migration on the dynamics of a prey–predator population. The functional response of the predator is governed by the Holling type II function. First, from the perspective of mathematical results, we develop results concerning the existence, uniqueness, positivity, boundedness, and dissipativity of solutions. Besides, many thresholds have been computed and used to investigate the local and global stability results by using the Routh–Hurwitz criterion and Lyapunov principle, respectively. We have also established the appearance of limit cycles resulting from the Hopf bifurcation. Numerical simulations are performed to explore the effect of migration on the dynamic of prey and predator populations.

Legendre wavelet method based solution of fractional order prey–predator model in type-2 fuzzy environment

Researchers have recently examined several fractional-order prey–predator population models in a crisp or type-1 fuzzy environment. But incompleteness in the membership may also be present. In this regard, type-2 fuzzy numbers are employed in this inquiry to address the system’s imprecision since they are more realistic than type-1 fuzzy numbers in light of the uncertain membership grades. To numerically estimate the solution, the Legendre wavelet method (LWM) is used in conjunction with the r2-cut of the r1-plane form of the type-2 fuzzy parameters. To test the applicability of the present method, three cases are examined. The present solutions are quantitatively compared with the RK4 solutions in the crisp case. Approximations of the type-2 fuzzy prey and predator populations are also included. In addition, two- and three-dimensional plots are depicted to examine the behaviour of prey and predator populations in crisp and type-2 fuzzy cases. To assess the LWM’s efficacy, comparison graphs of the acquired solutions and the results of the RK4 approach are also provided. Numerical validation against the RK4 method in crisp cases demonstrates strong agreement, while revealing contrasting population behaviours in first two scenarios, whereas in the third case, both populations reduce with time. Further, the behaviour of the fuzzy solution has also been shown graphically using the footprint of uncertainty (FOU).

Predator-induced prey dispersal can cause hump-shaped density-area relationships in prey populations.

Predation can both reduce prey abundance directly (through density-dependent effects) and indirectly through prey trait-mediated effects. Over the years, many studies have focused on describing the density-area relationship (DAR). However, the mechanisms responsible for the DAR are not well understood. Loss and fragmentation of habitats, owing to human activities, creates landscape-level spatial heterogeneity wherein patches of varying size, isolation and quality are separated by a human-modified "matrix" of varying degrees of hostility and has been a primary driver of species extinctions and declining biodiversity. How matrix hostility in combination with trait-mediated effects influence DAR, minimum patch size, and species coexistence remains an open question. In this paper, we employ a theoretical spatially explicit predator-prey population model built upon the reaction-diffusion framework to explore effects of predator-induced emigration (trait-mediated emigration) and matrix hostility on DAR, minimum patch size, and species coexistence. Our results show that when trait-mediated response strength is sufficiently strong, ranges of patch size emerge where a nonlinear hump-shaped prey DAR is predicted and other ranges where coexistence is not possible. In a conservation perspective, DAR is crucial not only in deciding whether we should have one large habitat patch or several-small (SLOSS), but for understanding the minimum patch size that can support a viable population. Our study lends more credence to the possibility that predators can alter prey DAR through predator-induced prey dispersal.

Stability, bifurcation, and control: Modeling interaction of the predator-prey system with Alles effect

In this study, we investigate the resilience and bifurcation properties of a discrete-time predator-prey model incorporating an Allee effect, the transfer of prey to predator energy, and a necessary threshold for prey consumption prior to predator reproduction. The discrete-time formulation involves iteratively adjusting the predator and prey populations at distinct time steps. The model encompasses equations that depict the fluctuations in population sizes, considering processes like breeding, predation, death, and the influence of the Allee effects. The investigation results indicate that the system's boundary equilibrium goes through a period-doubling bifurcation. Moreover, when the system is in close proximity to the exclusive positive equilibrium influenced by the prey-predator population, it exhibits Neimark-Sacker bifurcations. It's worth noting that our observations suggest that the predator population might approach extinction or achieve a stable equilibrium when chaotic dynamics appear in the prey population. The validity of this phenomenon is confirmed using the Lyapunov exponents calculation, which verifies the range of diverse dynamic behaviors. The outcomes and analysis of this research offer intriguing insights in both mathematical and biological realms. Implementing methods for chaos control could be beneficial in stabilizing or mitigating the occurrence of chaotic dynamics. Additionally, the theoretical results are supported by numerical simulations, phase portraits, bifurcation diagrams, graphs depicting chaos control, and maximum Lyapunov exponent graphs.

Analysis of Predator–Prey Model With Inclusion of Temperature Variability in Prey Refugees

One of the central concerns in ecology is assessing the impact of environmental changes, such as global temperature shifts, on the dynamic interactions between predators and prey across diverse ecosystems. This study explores the relationship between prey and predator populations in the context of climate‐induced temperature changes. Utilizing linearization and the Lyapunov approach, we analyze the local and global stability of the system’s equilibrium points. Applying Sotomayor’s theorem, we further investigate the local bifurcation of the fixed points. Additionally, the persistence of the model system is studied. The ecological models developed in this research aim to explore the stability of these predator–prey systems and provide insights to prevent the decline of endangered species. Numerical simulations are conducted to study the impact of environmental changes on the proposed models. The simulation results indicate that the temperature dependence of the predator–prey population dynamics is directly proportional to the size of the reserved habitat area for the prey when the temperature is high. These findings suggest that the preservation of sufficient habitat reserves plays a crucial role in promoting the long‐term survival of species and stabilizing their ecological interactions.

МОДЕЛИРОВАНИЕ ДИНАМИКИ ВЗАИМОДЕЙСТВУЮЩИХ ПОПУЛЯЦИЙ ТИПА «ХИЩНИК–ЖЕРТВА» ПРИ ПОСТОЯННОЙ МИГРАЦИИ ОСОБЕЙ С СОПРЕДЕЛЬНЫХ ТЕРРИТОРИЙ

The article deals with the research of the dynamics of a local predator-prey community with constant migration of individuals from neighboring territories. We studied several models with a constant inflow of individuals into both predator and prey populations. It is shown that changes in the overall dynamics are significantly influenced by the number of predator migrants: their large influx leads to the rapid and almost complete extinction of preys. The model considering only a constant prey inflow is successfully applied to modeling of various processes based on the principles of predator-prey population interactions, for example, when studying the consumption of difficult-to-renewable or nonrenewable natural resources. The study of the model provides allows getting estimations of the resources use efficiency and and the degree of modernization of the consumer sector. It is shown that two-dimensional models - modifications of the basic Volterra and Bazykin models, lead to structurally stable fluctuation regimes corresponding to the focus and limit cycle. We found that these models also contain fast-slow periodic dynamics, corresponding to a maximum limit cycle with strong spikes and smoother declines in the population size. The emergence of such regimes corresponds to the alternation of periods of active extinctions of prey, accompanied by an increase in the number of predators, and periods of long-term restoration of the prey, during which there is no harvesting of prey. We used methods of dynamic systems analysis to study the models. The construction of two-dimensional parametric portraits shows that, to get stable dynamics of the basic models of two interacting biological species, no complication is required, for example, by adding nonlinear terms. Stable regimes are also observed in simpler models, such as the original Lotka-Volterra or Bazikin models, where the recovery rate of prey population is a constant value.

Multi-scale dynamics of predator-prey systems with Holling-IV functional response

<abstract><p>In this paper, we propose a Holling-IV predator-prey system considering the perturbation of a slow-varying environmental capacity parameter. This study aims to address how the slowly varying environmental capacity parameter affects the behavior of the system. Based on bifurcation theory and the slow-fast analysis method, the critical condition for the Hopf bifurcation of the autonomous system is given. The oscillatory behavior of the system under different perturbation amplitudes is investigated, corresponding mechanism explanations are given, and it is found that the motion pattern of the non-autonomous system is closely related to the Hopf bifurcation and attractor types of the autonomous system. Meanwhile, there is a bifurcation hysteresis behavior of the system in bursting oscillations, and the bifurcation hysteresis mechanism of the system is analyzed by applying asymptotic theory, and its hysteresis time length is calculated. The final study found that the larger the perturbation amplitude, the longer the hysteresis time. These results can provide theoretical analyses for the prediction, regulation, and control of predator-prey populations.</p></abstract>

A stochastic model of predator-prey population dynamics

We present results of an analysis for a randomized three-dimensional predator-prey model representing the dynamics of wolf-deer interactions.

A predator–prey population dynamics simulation for biological control of Frankliniella occidentalis (Western Flower Thrips) by Orius laevigatus in strawberry plants

The use of predatory insect species to control agricultural pests is a key aspect of Integrated Pest Management (IPM) that can help to reduce the use of chemical pesticides. A simulation model that captures the population dynamics and interactions of this predator–prey system can serve as a valuable IPM decision-support tool. However, such a simulation is challenging to construct due to the interactions between life stages of insect predators, prey and the crop plants. Frankliniella occidentalis (Western Flower Thrips, WFT) is an economically devastating global pest that damages diverse crops. Orius laevigatus is an effective and commercially available predator that controls WFT. We, therefore, developed a population dynamics simulation that models their interactions to better understand these insects’ predator–prey dynamics in a strawberry crop. The model considers critical multi-stage predator–prey interaction, prey consumption reduction due to intraspecific competition, and the relationship between predation and oviposition rates of Orius. The simulation was structured using the boxcar train method, a stage-structured approach allowing insect populations to flow from one stage to the next over time. It was parameterised using data from existing literature. The results indicate the potential of the model to forecast the predator and prey species’ population dynamics for different initial populations. A global sensitivity analysis identified that the daily oviposition rate of WFT and egg survival rate of Orius were the most significant parameters. The prey handling time of Orius on WFT did not significantly impact population variation. This model provides a useful tool for scenario exploration and assessment of the effects of Orius on WFT that can aid decision-making in IPM prior to implementation.

Influence of multiple delays mechanisms on predator–prey model with Allee effect

Although there are many factors that influence population fluctuations, the impact of delay on population dynamics is one of the most significant ones. In this paper, we explore how the Allee effect in combination with a double delay (competition and cooperation) affects predator–prey population dynamics. One can determine the requirements for the equilibrium’s asymptotic stability and the presence of Hopf bifurcations by investigating the roots of a characteristic equation. The rigorous mathematical proofs of the stability and direction of the Hopf bifurcating periodic solutions are derived with the help of the normal form theory and the center manifold theorem. Numerical simulation shows that different delay mechanisms can generate different behaviors, such as the competitive delay that produces regular and irregular oscillations, as well as the chaotic, while the cooperative delay results in the stability switch. These theoretical and numerical results are illustrated that depending on the chosen delayed mechanism, delay may not necessarily lead to instability, and also can help for understanding the biological implications corresponding to the interaction dynamics of predator and prey.

A dynamic analysis of a prey–predator population model with a nonlinear harvesting rate

PurposeIn this article, the author discusses dynamical behaviors of a prey-predator population model with nonlinear harvesting rate and offers a mathematical analysis of the model.Design/methodology/approachThe design is by using modelization of populations interaction, qualitative theory of ordinary différential equations, bifurcations analysis, invariant center manifolds theory and Dulac's criterion.FindingsThe author studies the stability of solutions and the existence of periodic solutions in the model, and proves the existence of some invariant sets and the production of a transcritical together with a saddle-node bifurcation.Practical implicationsThe author studies the effects of harvesting on the persistence and extinction properties and its influence in the perspectives of economic views.Originality/valueThe authors considers a predator–prey model with a new nonlinear form of harvesting rate. The author’s intention is to make conceptual adjustments to a well-known predator–prey model in order to incorporate the effects of harvesting.

A fractal–fractional-order modified Predator–Prey mathematical model with immigrations

This manuscript aims to study a modified predator–prey model’s existence, stability, and dynamics under the newly developed fractal–fractional order operator in the Caputo–Fabrizio sense. The existence theory of the proposed model carries out through the Leray–Schauder alternative and sufficient conditions for stability are established using the classical technique of nonlinear functional analysis. The numerical results are obtained by the fractal–fractional Adam–Bashforth method in the Caputo–Fabrizio sense. The numerical results show that small immigrations invoke stable convergence in the predator–prey ecosystem. This means that a small number of sporadic immigrants can stabilize natural predator–prey populations.

DYNAMICS IN A FRACTIONAL ORDER PREDATOR–PREY MODEL INVOLVING MICHAELIS–MENTEN-TYPE FUNCTIONAL RESPONSE AND BOTH UNEQUAL DELAYS

The interrelationship between predator populations and prey populations is a central problem in biology and mathematics. Setting up appropriate predator–prey models to portray the development law of predator populations and prey populations has aroused widespread interest in many scholars. In this work, we propose a new fractional order predator–prey system involving Michaelis–Menten-type functional response and both unequal delays. Utilizing the contraction mapping theorem, we prove the existence and uniqueness of the solution to the considered fractional order predator–prey system. By virtue of some mathematical analysis techniques, nonnegativeness of the solution to the involved fractional order predator–prey system is analyzed. By constructing a suitable function, the boundedness of the solution to the considered fractional order predator–prey system is explored. Making use of Laplace transform, we derive the characteristic equation of the involved fractional order predator–prey system, then by means of the stability principle and the bifurcation theory of fractional order dynamical system, a series of novel delay-independent stability criteria and bifurcation conditions ensuring the stability of the equilibrium point and the creation of Hopf bifurcation of the considered fractional order predator–prey system, are built. The global stability of the involved fractional order predator–prey system is analyzed in detail. The role of time delay in controlling the stability and the creation of Hopf bifurcation is revealed. To check the legitimacy of the derived key results, software simulation results are effectively presented. The obtained results in this work are completely novel and play a significant role in maintaining ecological balance.

Analysis of a Holling-type IV stochastic prey-predator system with anti-predatory behavior and Lévy noise

<abstract><p>In this paper, we investigate a stochastic prey-predator model with Holling-type IV functional responses, anti-predatory behavior (referring to prey resistance to predator), gestation time delay of prey and Lévy noise. We investigate the existence and uniqueness of global positive solutions through Itô's formulation and Lyapunov's method. We also provide sufficient conditions for the persistence and extinction of prey-predator populations. Additionally, we examine the stability of the system distribution and validate our analytical findings through detailed numerical simulations. Our paper concludes with the implications of our results.</p></abstract>

The Stability of Ecosystems and Paradox of Enrichment via Spatial Prey-Predator Distributions with Active Dispersal

Understanding the transport mechanism is important key to interpret behavioral and physiological responses of organisms or to predict the success by which they find their resources. In this paper, the effect of transport mechanism in the prey-predator populations is investigated. The models suggested are based on classical Lotka-Volterra (LV) and Rosenzweig-MacArthur (RM) systems in which the passive dispersal (diffusion) and active dispersal (directed movement per se or taxis) are considered. The numerical simulation results obtained from this coupling model confirms two important features: 1) Active dispersal playing the convection role and considered as a simple mechanism to stabilize the prey-predator system; and 2) the coupling effect between active and passive dispersals generates heterogeneous distributions of both species of prey or of predator even in the homogenous environment.

Hydra effects predator-prey bazykin's model with stage-structure and intraspecific for predator

Bazykin's predator-prey population model is considered to represent the exchange stability condition of population growth. The existence of the hydra effect and, at the same time, analyzing its influence on population growth. The condition of the model divides the species into a stage structure, namely, prey, immature predators, and mature predators. The population growth of the three species has its own characteristics. This research revealed that the Holling type II and intraspecific predatory function responses together induce the Hydra effect. In the model formed, there are 12 equilibrium points, with details for every seven points of negative imaginary equilibrium and five points of non-negative equilibrium. The findings of research studies center on five points of non-negative equilibrium. All real roots that interpret the species population's growth conditions are taken and tested for long-term stability. The test results show one point of equilibrium that meets the Routh-Hurwitz criteria and their characteristic equations. In numerical simulations, the maximum sustained yield is in the local asymptotic stable state. The growth of prey trajectories increased significantly, although at the beginning of the interaction there was a slowdown in population growth. Meanwhile, the population of immature predators and mature predators was not significantly different. Both populations grow steadily toward the point of population stability. It turns out that the two populations grow inversely, the faster the rate of predation by predators, the faster the growth rate of the prey population.