Lotka-Volterra Research Articles

Stability and Hopf Bifurcation of a Delayed Predator–Prey Model with a Stage Structure for Generalist Predators and a Holling Type-II Functional Response

In this paper, we carry out some research on a predator–prey system with maturation delay, a stage structure for generalist predators and a Holling type-II functional response, which has already been proposed. First, for the delayed model, we obtain the conditions for the occurrence of stability switches of the positive equilibrium and possible Hopf bifurcation values owing to the growth of the value of the delay by applying the geometric criterion. It should be pointed out that when we suppose that the characteristic equation has a pair of imaginary roots λ=±iω(ω>0), we just need to consider iω(ω>0) due to the symmetry, which alleviates the computation requirements. Next, we investigate the nature of Hopf bifurcation. Finally, we conduct numerical simulations to verify the correctness of our findings.

Analysis of the Dynamical Properties of Discrete Predator-Prey Systems with Fear Effects and Refuges

This paper examines the dynamic behavior of a particular category of discrete predator-prey system that feature both fear effect and refuge, using both analytical and numerical methods. The critical coefficients and properties of bifurcating periodic solutions for Flip and Hopf bifurcations are computed using the center manifold theorem and bifurcation theory. Additionally, numerical simulations are employed to illustrate the bifurcation phenomenon and chaos characteristics. The results demonstrate that period-doubling and Hopf bifurcations are two typical routes to generate chaos, as evidenced by the calculation of the maximum Lyapunov exponents near the critical bifurcation points. Finally, a feedback control method is suggested, utilizing feedback of system states and perturbation of feedback parameters, to efficiently manage the bifurcations and chaotic attractors of the discrete predator-prey model.

Diffusion and distributed delay effects in a predator–prey system: A mathematical analysis

This study explores a nonautonomous model of prey-predator dynamics, featuring continuous distributed delay and diffusion, with time-dependent parameters. Our model depicts two ecological patches hosting two species: predators remain localized in one patch, while prey can disperse between both. Our analysis concentrates on ensuring the model’s consistent presence and achieving global stability. Additionally, we derive necessary conditions for periodic solutions by developing a pertinent Lyapunov function.

The effect of self-memory-based diffusion on a predator–prey model

The effect of self-memory-based diffusion on a predator–prey model

Study of bifurcation and delay-driven chaos in a prey-predator model with fear in prey reproduction and two forms of harvesting

Study of bifurcation and delay-driven chaos in a prey-predator model with fear in prey reproduction and two forms of harvesting

Dynamics Behavior of a Predatorg-Prey Diffusion Model Incorporating Hunting Cooperation and Predator-Taxis

In this paper, we consider a predator-prey diffusion model incorporating hunting cooperation and predator-taxis. Firstly, we establish the global existence of a classical solution for the model in any spatial dimension. Secondly, we analyze the stability/instability caused by predator-taxis, and we observe that predator-taxis play a key role in inducing stability changes. Specifically, if the positive equilibrium is stable for the corresponding reaction-diffusion model, the attractive predator-taxis can further stabilize the system, while the repulsive predator-taxis may lead to a change in spatial stability, if the positive equilibrium is unstable for the corresponding reaction-diffusion model, the attractive predator-taxis makes the model remain unstable, while the repulsive predator-taxis has a stabilizing effect. Finally, numerical simulations are employed to validate the obtained results.

An intuitionistic fuzzy approach for prey–predator harvesting system with toxicity and time delay

This study presents an intuitionistic fuzzy prey–predator model with time delay, where several toxic substances from external sources contaminate the prey and predator species. The system consists of one prey and one predator, with the logistic growth of prey species and predation terms proportional to prey density. This system incorporates Holling type-II functional response and introduces a time delay in harvesting prey species over a precise size or age. The primary focus of this work is to investigate the impact of age or size-selective delay on the system’s stability within the intuitionistic fuzzy environment. The model employs triangular intuitionistic fuzzy numbers to account for imprecise biological parameters, and it is converted into an intuitionistic fuzzy model using the Hukuhara derivative. The fuzziness of the model is then resolved using Yager’s Ranking method. Without delay, the paper delves into the system’s dynamics, examining local and global stability and bionomic considerations. Numerical simulations are performed using MATLAB to validate the theoretical findings.

Sediment DNA Records the Critical Transition of Bacterial Communities in the Arid Lake

It is necessary to predict the critical transition of lake ecosystems due to their abrupt, non-linear effects on social-economic systems. Given the promising application of paleolimnological archives to tracking the historical changes of lake ecosystems, it is speculated that they can also record the lake’s critical transition. We studied Lake Dali-Nor in the arid region of Inner Mongolia because of the profound shrinking the lake experienced between the 1300 s and the 1600 s. We reconstructed the succession of bacterial communities from a 140-cm-long sediment core at 4-cm intervals and detected the critical transition. Our results showed that the historical trajectory of bacterial communities from the 1200 s to the 2010s was divided into two alternative states: state1 from 1200 to 1300 s and state2 from 1400 to 2010s. Furthermore, in the late 1300 s, the appearance of a tipping point and critical slowing down implied the existence of a critical transition. By using a multi-decadal time series from the sedimentary core, with general Lotka-Volterra model simulations, local stability analysis found that bacterial communities were the most unstable as they approached the critical transition, suggesting that the collapse of stability triggers the community shift from an equilibrium state to another state. Furthermore, the most unstable community harbored the strongest antagonistic and mutualistic interactions, which may imply the detrimental role of interaction strength on community stability. Collectively, our study showed that sediment DNA can be used to detect the critical transition of lake ecosystems.

Bifurcation Analysis of a Class of Two-Delay Lotka–Volterra Predation Models with Coefficient-Dependent Delay

In this paper, a class of two-delay differential equations with coefficient-dependent delay is studied. The distribution of the roots of the eigenequation is discussed, and conditions for the stability of the internal equilibrium and the existence of Hopf bifurcation are obtained. Additionally, using the normal form method and the central manifold theory, the bifurcation direction and the stability for the periodic solution of Hopf bifurcation are calculated. Finally, the correctness of the theory is verified by numerical simulation.

IMPACT OF HUNTING COOPERATION AND FEAR EFFECT IN A GENERALIST PREDATOR–PREY MODEL

Predator foraging facilitation or hunting cooperation and the antipredator behavior of prey are essential mechanisms in evolutionary biology and ecology and may strongly influence the predator–prey dynamics. In a real-world scenario, this behavioral tendency is well documented, but less is known about how it could affect the dynamics between predator and prey. Here, we investigate the impact of the fear of predator on prey and the hunting cooperation in predator on the predator–prey dynamics, where the predator is assumed to be of generalist type. We observe that without fear, even with the high level of hunting cooperation, both populations may coexist, though the increasing level of hunting cooperation reduces the prey density at coexistence equilibrium. Moreover, increasing level of fear also destabilizes the system with and without hunting cooperation. Further, in the presence of hunting cooperation and fear effect, the model shows three different types of bistability phenomena: bistability between two coexisting equilibria, bistability between coexisting equilibria and prey-free equilibrium, and bistability between stable limit cycle and coexisting equilibria. In addition, saddle-node, Hopf, transcritical bifurcation of codimension one, Bautin (generalized Hopf), Bogdanov–Takens, and cusp bifurcation of codimension two are observed.

Global dynamics of a Lotka-Volterra competition-diffusion system with advection and nonlinear boundary conditions

Global dynamics of a Lotka-Volterra competition-diffusion system with advection and nonlinear boundary conditions

Impact of fear-induced group defense in a Monod–Haldane type prey–predator model

Impact of fear-induced group defense in a Monod–Haldane type prey–predator model

Methods for comparing theoretical models parameterized with field data using biological criteria and Sobol analysis

Prey–predator models are frequently developed to investigate trophic webs and to predict the population dynamics of prey and predators. However, the parameters of these models are often implemented without empirical data and may even be chosen arbitrarily. Commonly, only a few parameters are tested regarding their sensitivity and it is rare to read about the comparison between different prey–predator models (i.e. predation function structure). Here, we propose a method to compare four prey–predator models designed for two populations. We then apply this method to select the more biologically plausible one to model a simplified agricultural trophic system, including one predator compartment (the red fox Vulpes vulpes) and one prey group compartment (small mammals). These models are based on four Holling functional responses for the predation interaction and take the prey intrinsic growth rate into account through a Verhulst logistic function. Most parameters’ values (like attack rates or growth rates) were calculated from field data or based on literature review. We then used Sobol indices to conduct parameter exploration around mean parameter values to investigate and compare the model dynamics responses. Our first results showed that under our assumptions, the two most relevant models for our case study are the saturated Holling I and II models. We were also able to discriminate that among the 6 scaled parameters that vary, the model outputs are particularly sensitive to four of them (κ: saturation rate of the environment, Tr1: characteristic intrinsic decay time of the predators, Tc: characteristic growth time of the predator via the predation on the prey and λ: saturation rate of a predator’s stomach per time unit) and much less sensitive to two others (Tr2: characteristic intrinsic growth time of the prey and Ta: characteristic decay time of the prey due to the predation). These first encouraging results open the way for the next step, which will be to adapt this model construction to more complex prey–predator systems, with several predator and/or several prey compartments.

Spatio-temporal patterns and global bifurcation of a nonlinear cross-diffusion predator–prey model with prey-taxis and double Beddington–DeAngelis functional responses

The aim of this article is investigating the spatio-temporal patterns of a nonlinear cross-diffusion predator–prey model with prey-taxis and double Beddington–DeAngelis functional responses. First, by utilizing user-friendly version of Crandall–Rabinowitz bifurcation theory as an analytical method, the spatio-temporal patterns of positive steady state are obtained. Then, by regarding the cross-diffusion coefficient d1 as a bifurcation parameter, we will derive a bifurcation theorem for the corresponding nonlinear cross-diffusion model. Moreover, by applying spectrum theory, perturbation of simple eigenvalues and linearized stability, it is discovered that the bifurcation solutions possess local stability near the bifurcating point in proper conditions. These theoretical results mean that the cross-diffusion mechanism can create a coexistence environment for the preys and predator under some circumstances. Finally, a numerical example is proposed to verify our results.

Emergence of structure in plant–pollinator networks: low floral resource constrains network specialisation

Specialisation enhances the efficiency of plant–pollinator networks through the exchange of conspecific pollen transfer for floral resources. Floral resources form the currency of plant–pollinator interactions, but the understanding of how floral resources affect the structure of plant–pollinator networks remains modest. Previous theory predicts that optimally foraging animal species will specialise to improve resource acquisition under high resource availability. Although floral resource availability depends on both the plant production and animal consumption of the resources, previous work has assumed that production and availability are equivalent. This potentially may have led to erroneous inferences on the effect of resource availability on specialisation. We develop a mutualistic Lotka–Volterra consumer‐resource model to investigate the influence of floral resource availability on plant–pollinator network structure. The model incorporates animal adaptive foraging behaviour, floral resource dynamics, and density‐dependent dynamics. Specialisation, nestedness and modularity of simulated networks generated from the model under a wide range of parameters were explained using the generalised linear model. We found that the distinction between floral resource dynamics and plant density dynamics was necessary for partial specialisation of plant–pollinator networks. This is because floral resource dynamics constrained animal preference due to its depletion by animal species. Floral resource abundance had a positive effect on network specialisation, but animal density had a negative effect on network specialisation. Floral resource dynamics thus play key roles in the structure of plant–pollinator networks, distinctive from plant species density dynamics.

Stability, bifurcation, and chaos of a stage-structured predator-prey model under fear-induced and delay

The motivation of this paper is inspired by the research of Zanette et al. (2011) [4], which reveals that the perception of predation risk can reduce the number of juvenile prey. To investigate how fear affects the behavior of prey population, we formulate a predator-prey model with a stage structure for prey in which adult prey is undergoing the interference of fear. First, the dynamic properties of the model are analyzed, including the existence and stability of equilibria, bifurcations dynamics, such as transcritical, saddle-node, and Hopf bifurcations. Our results reveal that fear can promote the formation of the stability of the model by transforming it from an unstable state to a stable state. Numerical simulations and theoretical analysis indicated that when the level of fear reaches 25.4, the number of juvenile prey affected by fear decreases by 40% compared to those not affected by fear. Terrifyingly, the predators are on the brink of extinction in high levels of fear. Moreover, we examine the impact of time delay on fear in the previous model as prey needs some time to assess the predation risk. Some dynamic characteristics of this delayed model are also analyzed. Results indicate that the fear effect is conducive to generating a stable state, while delay is detrimental to generating a stable state, leading to limit cycles or even chaos.

Dynamics analysis of a nonlinear controlled predator–prey model with complex Poincaré map

In this paper, we propose a class of predator–prey models with nonlinear state-dependent feedback control in the saturated state. The nonlinear state impulse control leads to a diversity of pulse and phase sets such that the Poincaré map built on the corresponding phase sets behaves like the single-peak function and multi-peak function with multiple discontinuities. We start our study by analyzing the exact pulse and phase sets of models under various cases generated by the dependent parameter space of nonlinear state feedback control, then construct the Poincaré map that is followed by investigating their monotonicity, continuity, concavity, and immobility properties. We also explore the existence, uniqueness, and sufficient conditions for the global stability of the order-1 periodic solutions of the systems. Numerical simulations are carried out to illustrate and reveal the biological significance of our theoretical findings.

Disparate home range dynamics reflect nutritional inadequacies on summer range for a large herbivore

AbstractThe spatial distribution of animals has consequences for nutrition, predator–prey dynamics, spread of diseases, and population dynamics in general. Animals must establish a home range to secure adequate resources to fuel their energy needs. Home ranges, therefore, are temporally and spatially dynamic, given the changing requirements of an animal and the availability of resources on the landscape. We used data from two populations of bighorn sheep with contrasting population dynamics following pneumonia epizootics and different habitat quality on their summer range to test the hypothesis that the distribution and size of home ranges are influenced by environmental conditions and reproductive status. We used a combination of data from 768 vegetation transects and remotely sensed metrics to index forage quality of consecutive biweekly home ranges for 27 bighorn sheep, June–August 2019–2021. There were population differences in home range dynamics that were consistent with resource limitations in the population declining in abundance. Animals in both populations increased the size of their home range through the summer in association with declining forage quality indexed by plant phenology. Furthermore, animals in the Whiskey Mountain population without offspring had home ranges more than twice the size of animals with offspring, whereas there were no differences in the home range size between animals with and without offspring in Jackson. We demonstrated that limitations young offspring impose on space use of a mother may have consequences for animals living where larger home ranges are needed to secure adequate resources—sheep on Whiskey Mountain had to travel 1000 m from escape terrain to access the same amount of biomass that the Jackson sheep could access directly adjacent to escape terrain. Forage quality and availability influence movement and space use. In the presence of disease, movement and space use may influence pathogen transmission and persistence. Thus, forage availability may play an indirect role in population dynamics in the presence of disease, which is another line of evidence for how environmental and nutritional conditions may influence population dynamics when coping with disease.

Chaotic behavior in Lotka-Volterra and May-Leonard models of biodiversity.

Quantification of chaos is a challenging issue in complex dynamical systems. In this paper, we discuss the chaotic properties of generalized Lotka-Volterra and May-Leonard models of biodiversity, via the Hamming distance density. We identified chaotic behavior for different scenarios via the specific features of the Hamming distance and the method of q-exponential fitting. We also investigated the spatial autocorrelation length to find the corresponding characteristic length in terms of the number of species in each system. In particular, the results concerning the characteristic length are in good accordance with the study of the chaotic behavior implemented in this work.

Modeling the Complex Interplay: Dynamics of Job Displacement and Evolution of Artificial Intelligence in a Socio-Economic Landscape

Modeling the Complex Interplay: Dynamics of Job Displacement and Evolution of Artificial Intelligence in a Socio-Economic Landscape