Predator-prey Mathematical Model Research Articles

Dynamic analysis and optimal control strategies of a predator–prey mathematical model for the pest eradication in oil palm plantation

Oil palm cultivation stands as a crucial industry in Indonesia, significantly contributing to the nation’s economy by generating employment opportunities and fostering social welfare for communities residing near plantations. Despite its economic importance, oil palm plantations face various challenges, with one prominent issue being the infestation of nettle caterpillar pests. These pests cause leaf skeletonization, resulting in a staggering 36% reduction in oil palm productivity over a two-year period. This paper explores diverse strategies for pest management in oil palm plantations, encompassing biological control through the stimulation of natural predators, mechanical control involving the collection and incineration of cocoons, and chemical control through pesticide application. The research introduces a predator–prey mathematical model for oil palm plantation pests, where the leaf area serves as the primary food source for caterpillars, acting as prey. Through dynamic model analysis, four equilibrium points are identified, with interconnected conditions dictating their existence and stability. These conditions are visually represented in a bifurcation plane, providing concise information. The study further includes bifurcation diagrams of equilibrium points to elucidate the influence of each parameter on pests, predators, and the leaf area of oil palm plants. Additionally, sensitivity analysis of the stable interior equilibrium point is conducted to understand the impact of individual parameters. The paper extends its investigation to optimal control strategies, evaluating six scenarios categorized into two population conditions: with predators and without predators. Within each population condition, three control strategies are considered—chemical control only, mechanical control only, and a combination of chemical and mechanical control. Simulation results from the optimal control study reveal that the presence of natural predators emerges as a pivotal strategy in effectively managing nettle caterpillars. Notably, the control of resistant pests through pupa incineration has a substantial impact on reducing the caterpillar population in subsequent life cycles.

Prey–Predator Mathematics Model for Fisheries Insurance Calculations in the Search of Optimal Strategies for Inland Fisheries Management: A Systematic Literature Review

Fish stocking in inland fisheries involves a prey–predator interaction model so that the number of fish stocked affects optimal and sustainable yields. It is very important to make mathematical modeling to optimize inland fisheries management which is part of the blue economy. Currently, studies that focus on predator–prey mathematical modeling in inland fisheries, especially those related to insurance are lacking. The bibliometric database was taken from Google Scholar, Dimensions, Science Direct, and Scopus in the 2012–2022 research years. After further processing, it is displayed on the PRISMA diagram and visualized on VOSviewer to display the update of this research topic. As blue economy sustainability, the management of fisheries sector needs to be reviewed deeply. In this study, the assumptions of the predator–prey mathematical model are made to obtain the equilibrium point, maximum sustainable yield (MSY), and catch per unit effort (CPUE) values. These results can be used to calculate fisheries insurance as a strategy for optimizing sustainable fishermen’s income.

A predator-prey model of rice plant, sparrow, rat, and snake

Rice is one of the most widely consumed food sources in Indonesia. The success of the rice harvest can be influenced by several factors, one of which is plant pests. This can threaten national food security. Predator-prey mathematical model can be constructed to explain the relationship between rice plants with plant pests. The predator prey model consist of ODE system describing 2 predators level one (rats and sparrows), 1 predator level two (snake) and prey (rice plants). In this model, nine equilibrium points are obtained with analysis of the behavior of the model at each of its equilibrium points. We successfully simulated the model using hypothetical parameters and the results are agrees with the analysis behavior. Several factors will make plant pest populations decrease even lost from the population is the natural death and interaction of rice with plant pests

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.

PREDICTIVE CONTROL OF THE VARIABLE-ORDER FRACTIONAL CHAOTIC ECOLOGICAL SYSTEM

Since ecological systems are history-dependent, incorporating fractional calculus and especially variable order ones could significantly improve the emulation of these systems. Nonetheless, in the literature, no study considers ecological processes by variable-order fractional (VOF) model. This study is motivated by this issue. At first, we propose to extend a predator–prey mathematical model with VOF derivatives. The underlying assumption in the proposed model lies in considering values of fractional derivatives as time-varying functions instead of constant parameters. Some system’s dynamic features are investigated, and then the control of the proposed system is studied. To this end, a nonlinear model predictive control is offered for the VOF system. The necessary optimality and sufficient conditions for solving the nonlinear optimal control problem in the form of fractional calculus with variable-order derivative are formulated, and the controller’s design procedure is delineated. Finally, numerical simulations are performed to demonstrate the developed control technique’s effectiveness and performance for the VOF predator–prey model.

Analisis Kestabilan Model Matematika Predator-Prey pada Dinamika Sosial

In social life, difference and diversity is something that cannot be denied by anyone. Starting from differences horizontally concerning ethnicity, language, customs to religion and vertically concerning the political, social, cultural to economic fields. The existence of these many differences can certainly bring positive and negative impacts in social life. With diversity, interaction in society is dynamic, but it results in the emergence of negative attitudes such as egoism and competition between groups. From the occurrence of this can trigger the problem of social inequality in the community. Social inequality can occur because of national development efforts that only focus on economic aspects and forget about social aspects. The purpose of this thesis is to discuss the stability analysis of the predator-prey mathematical model on social dynamics with the Holling type II functional response. From this model analysis, we obtained four equilibrium points, which are the equilibrium point for the extinction of all population (E0) which is unstable, then the equilibrium point for the extinction of the non-poor population and the poor (E1) and the extinction of the non-poor population (E2) which are stable with certain conditions and coexistence (E3) which is to be asymptotically stable. Also in the final section, we perform the numerical simulation to supports the analytical result.

Reciprocal interactions between tumour cell populations enhance growth and reduce radiation sensitivity in prostate cancer

Intratumoural heterogeneity (ITH) contributes to local recurrence following radiotherapy in prostate cancer. Recent studies also show that ecological interactions between heterogeneous tumour cell populations can lead to resistance in chemotherapy. Here, we evaluated whether interactions between heterogenous populations could impact growth and response to radiotherapy in prostate cancer. Using mixed 3D cultures of parental and radioresistant populations from two prostate cancer cell lines and a predator-prey mathematical model to investigate various types of ecological interactions, we show that reciprocal interactions between heterogeneous populations enhance overall growth and reduce radiation sensitivity. The type of interaction influences the time of regrowth after radiation, and, at the population level, alters the survival and cell cycle of each population without eliminating either one. These interactions can arise from oxygen constraints and from cellular cross-talk that alter the tumour microenvironment. These findings suggest that ecological-type interactions are important in radiation response and could be targeted to reduce local recurrence.

Adaptation of species as response to climate change: Predator-prey mathematical model

Most of the species currently threatened with extinction seem to be under the pressure of unsuitable environmental conditions; e.g., climate change, scarce food resource, habitat fragmentation. One should expect species to have forms of resilience against such extinction. The point here is to examine the effect of spatial gradients on species survival against increasing temperature arising from climate change. Therefore, we start with the question of whether, when faced with extinction stemming from climate change, a spatial gradient and a beachhead have the power to prevent extinction. This problem is addressed theoretically using a coupled reaction diffusion equation for a predator-prey system in which the prey experiences an Allee effect. It is demonstrated that there exists a relationship between the slope of the gradient and the beachhead at which the predator-prey system can stably survive. The tendency of the system can be defined by a function where the system includes the threshold point for extinction, that separates the areas of extinction and survival. The findings reveal that spatial gradient can be used as a precaution, when the species faces to extinction, for species to create new habitat and sustain its persistence. Therefore, in this paper, it is shown that, in theory, the recovery of species from unsuitable environmental conditions can be achieved. This can be possible by taking into account the spatial gradient to slow down the forthcoming ecological extinction, and thus extend the system a while as an adaptation mechanism.

Dynamics of a fractional order predator-prey model with intraguild predation

ABSTRACTIn this paper, a fractional order predator–prey mathematical model has been developed. In this model, the concept of intraguild predation has been introduced. It is assumed that intermediate predator consumes only prey and intraguild predator consumes prey as well as intermediate predator. Also, intraspecific competition of prey, intermediate predator and intraguild predator has been considered. Uniqueness, boundedness and non-negativity of solutions of our proposed model have been discussed. Different possible equilibrium points are determined and the stability of our proposed model around these equilibrium points has been studied. It is found that fractional order system can show some interesting dynamics of our proposed model. Finally, some numerical simulation results have been presented to verify our theoretical findings.

Stability and dynamics of a fractional-order three-species predator-prey model.

In this paper, a fractional-order predator-prey mathematical model has been developed considering Holling type II functional response. Here, we have investigated the interaction dynamics of prey, middle predator and top predator. We assume that the middle predator consumes only the prey, and the top predator consumes only the middle predator. Here, the logistic growth of prey has been considered. Then, different possible equilibrium points of our proposed system are determined. The stability of our proposed system is investigated around the equilibrium points. Then, some numerical simulations results are presented for better understanding the dynamics of our proposed model. It is found that the fractional-order derivative can improve the stability of our proposed system.

Co-existence thresholds in the dynamics of the plant–herbivore interaction with Allee effect and harvest

In the interaction between plants and herbivores that live in the same ecosystem, understanding the conditions in which co-existence equilibrium occurs answers a major question in Ecology. In this interaction, plants serve as food for herbivores on the food chain. Then the livelihood of herbivores highly depends on the availability of food, in this case the availability of plants. Moreover, the abundance of the plant density alone does not guarantee the non-extinction of the herbivore population as they are assumed to reproduce sexually. With this motivation, in this paper a predator–prey mathematical model is reformulated such that the death rate of the herbivore population is dependent on the plant density and their emergence is also governed by the Allee effect. Using the mathematical theory of dynamical system, threshold conditions are obtained for the non-extinction of the herbivore population and a trapping region is obtained to ensure co-existence of the population. Moreover, it has been shown that the dynamics of the population is significantly sensitive to the feeding rate and the harvest rate of the herbivore population.

Optimal control of predator-prey mathematical model with infection and harvesting on prey

This paper presents a predator-prey mathematical model with infection and harvesting on prey. The infection and harvesting only occur on the prey population and it assumed that the prey infection would not infect predator population. We analysed the mathematical model of predator-prey with infection and harvesting in prey. Optimal control, which is a prevention of the prey infection, also applied in the model and denoted as U. The purpose of the control is to increase the susceptible prey. The analytical result showed that the model has five equilibriums, namely the extinction equilibrium (E0), the infection free and predator extinction equilibrium (E1), the infection free equilibrium (E2), the predator extinction equilibrium (E3), and the coexistence equilibrium (E4). The extinction equilibrium (E0) is not stable. The infection free and predator extinction equilibrium (E1), the infection free equilibrium (E2), also the predator extinction equilibrium (E3), are locally asymptotically stable with some certain conditions. The coexistence equilibrium (E4) tends to be locally asymptotically stable. Afterwards, by using the Maximum Pontryagin Principle, we obtained the existence of optimal control U. From numerical simulation, we can conclude that the control could increase the population of susceptible prey and decrease the infected prey.

Diffusion driven instability and Hopf bifurcation in spatial predator-prey model on a circular domain

In this paper, we investigate theoretically and numerically a 2-D spatio-temporal dynamics of a predator-prey mathematical model which incorporates the Holling type II and a modified Leslie–Gower functional response and logistic growth of the prey. This system is modeled by a reaction diffusion equations defined on a disc domain {(x,y)∈R2/x2+y2<R2} with Dirichlet initial conditions and Neumann boundary conditions. We study the local and global stability of the positive equilibrium point. We show that the diffusion can induce instability of the uniform equilibrium point which is stable with respect to a constant perturbation as shown by Turing in 1950s and derive the conditions for Hopf and Turing bifurcation in the spatial domain. Numerical results are given in order to illustrate how biological processes affect spatiotemporal pattern formation in a spatial domain. We perform the computations and generalize, on a circular domain, the results presented in Camara and Aziz-Alaoui [6].

A mathematical study of a predator–prey model with disease circulating in the both populations

Disease in ecological systems plays an important role. In the present investigation we propose and analyze a predator–prey mathematical model in which both species are affected by infectious disease. The parasite is transmitted directly (by contact) within the prey population and indirectly (by consumption of infected prey) within the predator population. We derive biologically feasible and insightful quantities in terms of ecological as well as epidemiological reproduction numbers that allow us to describe the dynamics of the proposed system. Our observations indicate that predator–prey system is stable without disease but high infection rate drive the predator population toward extinction. We also observe that predation of vulnerable infected prey makes the disease to eradicate into the community composition of the model system. Local stability analysis of the interior equilibrium point near the disease-free equilibrium point is worked out. To study the global dynamics of the system, numerical simulations are performed. Our simulation results show that for higher values of the force of infection in the prey population the predator population goes to extinction. Our numerical analysis reveals that predation rates specially on susceptible prey population and recovery of infective predator play crucial role for preventing the extinction of the susceptible predator and disease propagation.

Coexistence and optimal control problems for a degenerate predator–prey model

In this paper we present a predator–prey mathematical model for two biological populations which dislike crowding. The model consists of a system of two degenerate parabolic equations with nonlocal terms and drifts. We provide conditions on the system ensuring the periodic coexistence, namely the existence of two non-trivial non-negative periodic solutions representing the densities of the two populations. We assume that the predator species is harvested if its density exceeds a given threshold. A minimization problem for a cost functional associated with this process and with some other significant parameters of the model is also considered.

A predator–prey mathematical model with both the populations affected by diseases

A prey–predator model with disease in both the population is proposed and analyzed. Local stability analysis of biologically feasible equilibrium points is worked out with help of ecological as well as disease basic reproduction numbers. We also explain the community composition of ecosystem by basic reproduction numbers. It is shown that the system is globally stable around the interior equilibrium point under certain parametric conditions. Moreover, the system is persistent under some feasible parametric conditions. Our analysis reveals that force of infection and predation rate are the key parameters on the dynamics of the system. Extensive numerical experiments with hypothetical set of parameters are carried out to substantiate our analytical findings. The oscillatory coexistence of the species which is very common in nature is observed for predator disease free system. The introduction of disease in the predator population enhances the extinction of predator population. We also observe that the system will be disease free and all population will persist through proper predational strategies.