Lotka-Volterra Research Articles

Dynamics of a slow-fast Leslie-Gower predator-prey model with prey harvesting.

For the Leslie-Gower predator-prey model with Michaelis-Menten type prey harvesting, the known results are on the saddle-node bifurcation and the Hopf bifurcation of codimensions 1, the Bogdanov-Takens bifurcations of codimensions 2 and 3, and on the cyclicity of singular slow-fast cycles. Here, we focus on the global dynamics of the model in the slow-fast setting and obtain much richer dynamical phenomena than the existing ones, such as global stability of an equilibrium; an unstable canard cycle exploding to a homoclinic loop; coexistence of a stable canard cycle and an inner unstable homoclinic loop; and, consequently, coexistence of two canard cycles: a canard explosion via canard cycles without a head, canard cycles with a short head and a beard and a relaxation oscillation with a short beard. This last one should be a new dynamical phenomenon. Numerical simulations are provided to illustrate these theoretical results.

Population dynamics in a Leslie - Gower predator-prey model with proportional prey refuge at low densities

In this paper we propose a mathematical Leslie-Gower predator-prey model, in which the prey takes refuge from the predator when its population size is below a critical threshold, the functional response of the predator is represented by a Holling II function, and the growth of the prey in the absence of the predator is subject to a semi-saturation parameter that affects its birth curve. Since the model is composed of two vector fields, its qualitative analysis includes, in addition to the determination of the number and stability of the equilibria for each vector field and belonging to the biological sense set, the study of the dynamics in the trajectories close to the dividing curve of the two vector fields in order to determine possible pseudo-equilibria. As a result, if the proposed model has a single inner equilibrium, then there is the possibility of having between one or at least two limit cycles, coexisting or not in both vector fields and around the inner equilibrium. Likewise, the model has a stable pseudo-equilibrium which may be surrounded by at least two limit cycle.

COMPLEX DYNAMICAL BEHAVIORS OF A DISCRETE MODIFIED LESLIE–GOWER PREDATOR–PREY MODEL WITH PREY HARVESTING

Recently, the research on the modified Leslie–Gower model has become an appealing topic. Due to economic efficiency and the complexity of discrete models, we investigate a discrete modified Leslie–Gower predator–prey model with prey harvesting in this paper. The stability of fixed points and bifurcations of the interior fixed points are studied. According to bifurcation theory and normal forms, we derived the conditions of codimension 2 bifurcations occurred, including 1:1 strong resonance bifurcation and fold-flip bifurcation. These two bifurcations are unusual in bifurcation analysis on discrete systems. In addition, the continuation curves, bifurcation diagrams, and phase diagrams are used to demonstrate theoretical results. Our study shows the interesting dynamics of this model that are very different from the continuous one.

Numerical Study of the Reaction Diffusion Prey–Predator Model Having Holling II Increasing Function in the Predator Under Noisy Environment

The current study deals with the stochastic diffusive prey–predator model with Holling II increasing function in the predator. As it is a population dynamics its population can be affected by various factors such as disease in prey–predator species, environmental fluctuation, etc. So, the underlying model is considered under the impact of a temporal multiplicative noise. The numerical solution of the governing model is obtained by finite difference schemes. To check the accuracy of the numerical solution, the stability and consistency of the schemes in the mean square sense are derived. All the schemes are consistent with the model and von Neumann stable. Several simulations are drawn to check the efficacy of the scheme. A test problem is investigated with numerical schemes and for the different values of the parameters, 3D and 2D plots are drawn. A graphical representation of the given system is drawn which shows the behavior of both species and the efficiency of the novel schemes. Hope so, these results will motivate the researcher to consider the stochastic prey–predator model for a better understanding of the ecosystem.

An empirical predator‒prey model: Examining the interaction of Florida panthers, white‐tailed deer, and cattle in Florida

AbstractIn this article, we present a predator‒prey model for Florida panthers (FL panthers), white‐tailed deer (deer) and cattle in Florida. The model is used to construct the following four policy scenarios: (a) joint management of all three species (Scenario 1); (b) a target population size of FL panthers of 240 individuals (Scenario 2); (c) a target population size of FL panthers equal to the carrying capacity (Scenario 3); and (d) a target population size of FL panthers of zero (Scenario 4) (Scenario 4). We parameterized the model for FL panthers, deer and cattle in Florida, which enables us to obtain empirical results for the population, harvest and net benefit of all three species under each policy scenario. When comparing Scenarios 1 and 2, the total net benefit of all three species is only slightly higher in the former case. Under Scenarios 3 and 4, the net benefit of deer changes significantly compared to Scenario 1, while the effect on the net benefit of cattle is minor. When comparing Scenario 1 and an actual situation, we find that the optimal population of FL panthers is only slightly higher than the actual population.

Role of eggplant trichome in whitefly oviposition and its relevance to biological control under greenhouse conditions.

The combined release of the predatory mite Amblyseius swirskii (Athias-Henriot) and the mirid Nesidiocoris tenuis (Reuter) provides effective biological control of the tobacco whitefly (Bemisia tabaci (Gennadius)) in greenhouse eggplant. However, knowing how plants' trichomes affect pest-predator interactions could improve whitefly management. Here, the effect of two varieties with either the presence or absence of trichomes was assessed on naturally occurring whitefly populations and predator abundance in a first experiment under field conditions. Predator-prey models were developed to assess the effect of trichomes on pest and predator population dynamics under field conditions. In a second semi-field experiment, the occurrence and oviposition preferences of B. tabaci and A. swirskii in the same eggplant varieties were compared. Significantly higher numbers of whitefly and mite, adults and eggs, were found on the hairy variety in both experiments. However, no differences were found in N. tenuis abundance between varieties under field conditions. Predator-prey models showed that whitefly growth rate increased in the hairy variety. N. tenuis and A. swirskii showed different fitness parameters according to the variety, with the former displaying better performance in the hairless variety and the latter in the hairy variety. Both predators effectively controlled the increase in whitefly populations in both varieties. Overall, the findings suggest that the hairless variety is more effective in deterring whiteflies. Additionally, the higher population of A. swirskii on the hairy variety indicates that this predator benefited from both the presence of trichomes and the prey.

Traveling front in a diffusive predator–prey model with Beddington–DeAngelis functional response

In this paper, we consider a singular diffusive predator–prey model with Beddington–DeAngelis functional response, employing geometric singular perturbation theory and Bendixson's criteria. Our investigation revolves around transforming the reaction–diffusion equation into a multi‐scale four‐dimensional slow–fast system with two different orders of small parameters. Through once singular perturbation analysis, our focus shifts towards exploring the existence of heteroclinic orbits in a three‐dimensional system. We analyze these dynamics through the perspective of the Fisher–KPP equation in two limit cases. In the first case, only the normal to the two‐dimensional slow manifold is unstable. This allows for the deduction of existence of heteroclinic orbits in the three‐dimensional system through investigating the dynamics on the two‐dimensional slow manifold. Consequently, we obtain both monotonic traveling fronts and non‐monotonic fronts with oscillatory tails. In the second case, the normal to the one‐dimensional slow manifold exhibits both stable and unstable directions, then it is impossible to restrict the dynamics of the three‐dimensional system entirely to the slow manifold. Instead, we integrate the slow orbits of the reduced system with the fast orbits of the layer system to construct a singular heteroclinic orbit. According to Fenichel's theorem, we discover the existence of exact heteroclinic orbits of three‐dimensional system and derive the monotonic traveling fronts under weaker parameter conditions. Additionally, we also discuss the nonexistence of traveling fronts. Finally, we demonstrate our theoretical results with numerical simulations.

First Integrals and a Zero-Hopf Bifurcation of the Four-Dimensional Lotka-Volterra Systems

In this paper, the integrability and a zero-Hopf bifurcation of the four-dimensional Lotka-Volterra systems are studied. The requirements for this kind of system's integrability and a line of singularities with two zero eigenvalues are provided. We identify the parameters that lead to a zero-Hopf equilibrium point at each point along the line of singularities. We show that there is only one parameter that displays such equilibria. The first-order averaging method is also employed, although this method will not give any information about the bifurcate periodic solutions that bifurcate from the zero-Hopf equilibria.

Properties of Cubic Fuzzy Competition Graphs with Applications

This study extends the concept of competition graphs to cubic fuzzy competition graphs by introducing additional variations including cubic fuzzy out-neighbourhoods, cubic fuzzy in-neighbourhoods, open neighbourhood cubic fuzzy graphs, closed neighbourhood cubic fuzzy graphs, cubic fuzzy (k) neighbourhood graphs and cubic fuzzy [k]-neighbourhood graphs. These variations provide further insights into the relationships and competition within the graph structure, each with its own defined characteristics and examples. These cubic fuzzy CMGs are further classified as cubic fuzzy k-competition graphs that show competition in the k th order between components, p -competition cubic fuzzy graphs that concentrate on competition in terms of membership degrees, and m -step cubic fuzzy competition graphs that analyze competition in terms of steps. Further, some related results about independent strong vertices and edges have been obtained for these cubic fuzzy competition graph classes. Finally, the proposed concept of cubic fuzzy competition graphs is supported by a numerical example. This example showcases how the framework of cubic fuzzy competition graphs can be practically applied to the predator-prey model to illustrate the representation and analysis of ambiguous information within the graph structures.

Dynamics of a Delayed Prey-Predator System With Crowley-Martin Functional Response

ln this paper, an ecological predator-prey model with time delay is proposed and studied analytically and numerically. The stability and bifurcation analysis of the proposed model with a Crowley-Martin response function are covered in this article. In the beginning, equilibrium points of the proposed model are identified. Secondly, the local asymptotic stability of the equilibria and Hopf-bifurcation are discussed by the characteristic equations of the system. Finally, by creating a suitable Lyapunov function and LaSalle’s invariance principle, the equilibria's global asymptotic stability is examined.

Dynamic Analysis of a Fractional-Modified Leslie-Gower Model with Predator Harvesting

This paper presents an investigation of a modified Leslie-Gower predator-prey model that incorporates fractional discrete-time Michaelis-Menten type prey harvesting. The analysis focuses on the topology of nonnegative interior fixed points, including their existence and stability dynamics. We derive conditions for the occurrence of flip and Neimark-Sacker bifurcations using the center manifold theorem and bifurcation theory. Numerical simulations, conducted with a computer package, are presented to demonstrate the consistency of the theoretical findings. Overall, our study sheds light on the complex dynamics that arise in this model and highlights the importance of considering fractional calculus in predator-prey systems with harvesting.

Long-term monitoring of a population of greater horseshoe bat emphasises the importance of a pest beetle prey on demographic trends.

The global decline in insect biomass has far-reaching implications for terrestrial and freshwater food webs, impacting species reliant on insects as a crucial component of their diet. This issue extends to species traditionally considered agricultural pests, such as the common cockchafer Melolontha melolontha. In the race to combat cockchafers through collection, insecticide use, and other control methods, the repercussions of their numerical fluctuations on predators, including species of high conservation importance like bats, have been largely overlooked. Drawing on 31-years of monitoring data for a greater horseshoe bat Rhinolophus ferrumequinum population in the Aosta Valley (Western Italian Alps), we investigated whether annual fluctuations in bat counts are influenced by cockchafer availability and weather conditions. Despite an overall positive trend in bat abundance, we observed pronounced annual fluctuations, mostly driven by cockchafer availability rather than variations in temperature and precipitation. Furthermore, we found a significant association between cockchafer availability and the median date of birth and birth rate of bats. Births occurred approximately 5 days earlier in cockchafer flight years, with earlier births also linked to warmer spring temperatures and higher numbers of warm days in April. Moreover, the ratio pups/older bats was 0.56 in cockchafer flight years, compared to 0.47 in other years. Our results underscore the importance of considering predator-prey dynamics when examining the long-term population trends of species of conservation concern. We recommend implementing restrictions on the use of chemicals and other potentially harmful practices that may diminish prey abundance or quality, including that of species considered as agricultural pests. In designing conservation strategies, a delicate balance should be struck between the current interests of farmers and the overarching goal of preserving biodiversity against potential future threats.

The eco-evolutionary dynamics of two strategic species: From the predator-prey to the innocent-spreader rumor model

Species frequently engage in both competitive and cooperative interactions, delicately balancing these dynamics to optimize their chances of survival and reproduction. While competition drives individuals to compete for limited resources, cooperation can emerge as a strategic response, mitigating risk and enhancing collective payoff. To bridge theoretical game approaches such as payoff, cooperation, and defections in ecological systems, we propose a two-species predator–prey model inspired by the principles and variations of the prisoner’s dilemma game. We comprehensively address and analytically verify all stable strategic states, exploring the role of payoff parameters both individually and collectively. Additionally, we investigate the effect of free space. Beyond ecological contexts, we present a model of rumor propagation within a social system to establish connections with the prisoner’s dilemma game. In both systems, our primary focus is to discuss strategies and enhance the cooperative factor within the system, given its crucial importance across diverse environments.

Symbolic-numeric algorithm for parameter estimation in discrete-time models with exp

Dynamic models describe phenomena across scientific disciplines, yet to make these models useful in application the unknown parameter values of the models must be determined. Discrete-time dynamic models are widely used to model biological processes, but it is often difficult to determine these parameters. In this paper, we propose a symbolic-numeric approach for parameter estimation in discrete-time models that involve univariate non-algebraic (locally) analytic functions such as exp. We illustrate the performance (precision) of our approach by applying our approach to two archetypal discrete-time models in biology (the flour beetle ‘LPA’ model and discrete Lotka-Volterra competition model). Unlike optimization-based methods, our algorithm guarantees to find all solutions of the parameter values up to a specified precision given time-series data for the measured variables provided that there are finitely many parameter values that fit the data and that the used polynomial system solver can find all roots of the associated polynomial system with interval coefficients.

Analysis of impacts on larger ecosystems resulting from dynamic sex ratio of lampreys based on Lotka-volterra and Predator model

Analysis of impacts on larger ecosystems resulting from dynamic sex ratio of lampreys based on Lotka-volterra and Predator model

Prey-Predator model with optimal harvesting in fuzzy environment

Prey-Predator model with optimal harvesting in fuzzy environment

Rich dynamics of a reaction–diffusion Filippov Leslie–Gower predator–prey model with time delay and discontinuous harvesting

To reflect the harvesting effect, a nonsmooth Filippov Leslie–Gower predator–prey model is proposed. Unlike traditional Filippov models, the time delay and reaction–diffusion under the condition of homogeneous Neumann boundary are considered in our system. The stability of equilibrium and the existence of the spatial Hopf bifurcation of the subsystems at the positive equilibrium are investigated. Furthermore, a comprehensive analysis is conducted on the sliding mode dynamics as well as the regular, virtual, and pseudoequilibria. The findings reveal that our Filippov system exhibits either a globally asymptotically stable regular equilibrium, a globally asymptotically stable time periodic solution, or a globally asymptotically stable pseudoequilibrium, contingent upon the specific values of the time delay and threshold level. A boundary point bifurcation, which transform a stable equilibrium point or periodic solution into a stable pseudoequilibrium, is demonstrated to emphasize the impact of time delay on our Filippov system and the significance of threshold control. Meanwhile, two kinds of global sliding bifurcations are exhibited, which sequentially transform a stable periodic solutions below the threshold into a grazing, sliding switching, and crossing bifurcations, depending on changes in the time delay or threshold level. Our results indicate that bucking bifurcation and crossing bifurcation pose significant challenges to the control of our Filippov system.

Data‐driven control and a prey–predator model for sourcing decisions in the low‐carbon intertwined supply chain

AbstractThis paper addresses the challenges of low‐carbon sourcing in intertwined supply chains by proposing a data‐driven control framework and a prey–predator model for sourcing decisions. The objective is to optimize low‐carbon objectives and reduce environmental impact. Existing static models fail to capture the dynamic nature of supply chain systems and overlook the ripple effects when sourcing decisions propagate throughout the interconnected network. To bridge this gap, our study develops a dynamic model that explicitly captures the bullwhip effect and leverages real‐time and historical data. This model conceptualizes suppliers as prey and manufacturers and consumers as predators, employing an ecological analogy to decipher the intricate interactions and dependencies within the supply chain. Through this approach, we identify strategies to promote sustainable practices and motivate suppliers to adopt low‐carbon measures. We assess two data‐driven algorithms, the nonlinear auto‐regressive exogenous (NARX) network and sparse identification of nonlinear dynamic systems with input variables (SINDYc). The results reveal that SINDYc outperforms prediction accuracy and control, offering significant advantages for rapid decision‐making. The study highlights how shifts in market demands and regulatory pressures critically influence the strategies of chemical firms and fertilizer markets. Moreover, it discusses the economic challenges in transitioning from high carbon footprint suppliers (HCFSs) to low carbon footprint suppliers (LCFSs), exacerbated by a notable cost disparity where HCFSs are approximately 30% cheaper. By advancing beyond conventional static models, this research provides a deeper understanding of the environmental impacts and operational dynamics within supply chains, emphasizing the significant “ripple effect” where decisions at one node profoundly affect others within the chain.

Dynamics of a controlled discrete prey-predator model with Holling type-II functional response

Dynamics of a controlled discrete prey-predator model with Holling type-II functional response

Pattern dynamics in a predator–prey model with Smith growth function and prey refuge in predator poisoned environment

In this article, we investigate temporal as well as spatiotemporal dynamics of a predator–prey system where prey grows according to Smith growth function and prey refuges in predator poisoned environment. Self as well as cross diffusion are taken into consideration to describe spatial moment of the species which make the system more realistic. This study aims to investigate the role of prey refuge, toxin-carrying carcass and the impact of cross-diffusion on the spatial distribution of species densities and associated patterns. The existence of equilibria and their stability conditions for non-spatial system are derived. We discuss the occurrence of Hopf bifurcation and detect the stability of limit cycle by computing first Lyapunov number. Temporal dynamics of the system is studied by varying the significant parameters. Study reveals that both prey refuge and fatality rate of predator due to toxin-carrying carcass, initially makes the system stable for low refuge and fatality rate but predator population goes to extinction as those effects increase. Turing instability conditions are listed and different instability regions so formed are discussed. In Turing region, pattern selection with the help of amplitude equation via weakly nonlinear analysis are performed which also numerically supported. Intensity of refuge and fatality rate due to toxin carrying carcass result paradox with real ecological world due to the presence of cross diffusion. Various kind of patterns such as mixture of stripe and spot, spot, irregular chaotic patterns emerge due to Hopf and Turing instability. Particularly, spatiotemporal chaos is a controversial topic because of its significant consequences for population dynamics. The results of this work are expected to contribute to a better understanding the level of prey refuge, use of toxin-carrying carcass considering movement in population and effect of cross-diffusion.