Predator Model Research Articles

Pattern formation and delay-induced instability in a Leslie–Gower type prey–predator system with Smith growth function

In this article, we have investigated the spatiotemporal dynamics and delay-induced instability of a Leslie–Gower type prey–predator model under the influence of environmental toxicants with Smith growth function. This growth function is more realistic than logistic growth as it better describes the growth of the biological population. It has been used where the growth limitations are based on the proportion of available resources not utilized. A few works of Smith’s growth models are reported in the literature. Therefore, spatiotemporal dynamics and pattern formation with delay effect remain an exciting area of research, which motivates the present work. This work has studied two types of dynamical systems: (i) an ordinary differential temporal system with time delay and (ii) a reaction–diffusion system with time delay. The existence of equilibrium points and their stability conditions are discussed. Hopf bifurcation emerges in both proposed systems with respect to delay parameter. The stability and direction of Hopf bifurcation and delay–diffusion-driven instability have been investigated for the reaction–diffusion system. Numerical simulation is performed to support the analytical results and theorems. Moreover, the existence of Hopf and delay-induced instability are proved numerically. Interesting one-dimensional regular and irregular stripe patterns are obtained for increased values of the time delay parameter. Also, the presence of natural toxicants has a negative impact on the growth of prey–predator species.

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.

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 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

Dynamical behaviors of autonomous and nonautonomous models of generalist predator–prey system with fear, mutual interference and nonlinear harvesting

In this study, we explore a predator–prey model to evaluate the impacts of fear, mutual interference, and harvesting, as these factors play crucial roles in modifying population dynamics, trophic cascades, and species interactions. Additionally, we enhance the model to incorporate the seasonality of specific ecological factors across time. The autonomous and associated nonautonomous models have undergone thorough mathematical analysis and extensive numerical simulations in the course of the study. We conduct a fundamental differential sensitivity analysis to discern the key parameters influencing both prey and predator populations within the ecosystem. Through numerical investigations of the autonomous model, we notice that in the presence (absence) of harvesting, the mutual interference among predators destabilizes (stabilizes) the coexisting equilibrium. The fear effect significantly reduces the densities of prey and predator populations, and destabilizes the coexisting equilibrium by producing multiple stability switches. The survival of predators substantially depends on the focal prey if their growth from other food sources is relatively low. Notably, the harvesting of predators is beneficial and promotes the coexistence of species only when their growth from other food sources is too much. Moreover, the system shows bistability, tristability, and multiple limit cycles under identical ecological conditions. We also show various phenomena that manifest within the seasonally forced model, including the emergence of periodic solutions, higher periodic solutions, bursting patterns, and chaotic dynamics. Remarkably, the influence of seasonal forcing possesses the capability to govern the chaotic dynamics by giving rise to an exceedingly quasi-periodic arrangement in the prey and predator densities.

On wavefront patterns in a fractional reaction–diffusion model for predator–prey system with anti-predator behavior

AbstractIn recent studies, much attention has been paid to reaction–diffusion systems with anomalous diffusion. In this paper, we investigate the formation of wavefront patterns in a predator–prey model with anti-predator behavior under the influence of anomalous subdiffusion. We use methods of traveling wave analysis and numerical integration to establish the existence of traveling wavefront solutions. Further, obtained traveling wavefront solutions are validated through direct computer simulations of time-dependent solutions for fractional partial differential equation system. It is found that wavefronts exist in a range of system parameters, which travel faster in the subdiffusive system than in the normal diffusive one.

Impact of wind flow and global warming in the dynamics of prey–predator model

Prey–predator interaction is among the most crucial aspects of ecology. In this paper, the dynamics of a prey–predator system in which the prey and predator populations experience a birth rate reduction due to global warming and the wind flow effect on them are investigated. The system’s well-posedness before presenting findings about the existence of potential equilibrium points is studied. Conditions under which a different number of interior equilibria exist are also derived here. It is very important to know that when the positive equilibrium point is stable, this reflects positively on the ecosystem, its continuity and persistence. The conditions that have been identified are conditions that preserve the ecosystem from extinction when environmental influences such as global warming and wind speed are present. In addition, the local stability study around all equilibrium points and established the transversality criteria for the occurrence of transcritical bifurcations and Hopf bifurcations is conducted. The existence of different equilibrium points numerically by altering the prey’s global warming effect is illustrated , also showcasing the model’s dynamic character from a mathematical perspective. For significant amounts of global warming, its notice that the system exhibits periodic behavior. Thus, ecologists can approximate the parameters necessary to investigate and obtain such crucial findings regarding nonlinear systems with the aid of the current study.

Optimal control of an infected prey–predator model with fear effect

In this paper, we propose and analyze a prey–predator model with the functional response of Beddington–DeAngelis and the fear effect that have infection only in prey populations. We determine existence criteria of several equilibria, and the stability at different equilibria are presented. We exert pesticide control over prey and additional food control over predators, the optimal control is obtained by the Pontryagin maximum principle. We confirm that adding controls to the predator and prey yields better results. Further we enrich our analysis with the inclusion of the existence and uniqueness of the optimal control. Finally, some numerical results to illustrate our analysis are presented.

Does mutual interference stabilize prey–predator model with Bazykin–Crowley–Martin trophic function?

We investigated a system of ordinary differential equations that describes the dynamics of prey and predator populations, taking into account the Allee effect affecting the reproduction of the predator population, and mutual interference amongst predators, which is modeled with the Bazykin–Crowley–Martin (BCM) trophic function. Bifurcation analysis revealed a rich spectrum of bifurcations occurring in the system. In particular, analytical conditions for the saddle–node, Hopf, cusp, and Bogdanov–Takens bifurcations were derived for the model parameters, quantifying the strength of the predator interference, the Allee effect, and the predation efficiency. Numerical simulations verify and illustrate the analytical findings. The main purpose of the study was to test whether the mutual interference in the model with BCM trophic function provides a stabilizing or destabilizing effect on the system dynamics. The obtained results suggest that the model demonstrates qualitatively the same pattern concerning varying the interference strength as other predator-dependent models: both low and very high interference levels increase the risk of predator extinction, while moderate interference has a favorable effect on the stability and resilience of the prey–predator system.

Spatiotemporal and Trade-Off Dynamics in Prey–Predator Model with Domed Functional Response and Fear Effect

In the ecological scenario, predators often risk their lives pursuing dangerous prey, potentially reducing their chances of survival due to injuries. Prey, on the other hand, try to strike a balance between reproduction rates and safety. In our study, we introduce a two-dimensional prey–predator model inspired by Tostowaryk’s work, specifically focusing on the domed-shaped functional response observed in interactions between pentatomid predators and neo-diprionid sawfly larvae. To account for the varying effectiveness of larval group defense, we incorporate a new component into the response equation. Our investigation delves into predator trade-off dynamics by adjusting the predator’s mortality rate to reflect losses incurred during encounters with dangerous prey and prey’s trade-off between safety and reproduction rate incorporating this domed-shaped functional response. Our model demonstrates bistability and undergoes various bifurcations, including transcritical, saddle-node, Hopf, Bogdanov–Takens, and Homoclinic bifurcations. Critical parameters impact both predator and prey populations, potentially leading to predator extinction if losses due to dangerous prey encounters become excessive, highlighting the risks predators face for their survival. Furthermore, the efficacy of group defense mechanisms can further endanger predators. Expanding our analysis to a spatially extended model under different perturbations, we explore Turing instability to explain the relationship between diffusion and encounter parameters through both stationary and dynamic pattern formation. Sensitivity to initial conditions uncovers spatiotemporal chaos. These findings provide valuable insights into comprehending the intricate dynamics of prey–predator interactions within ecological systems.

Study of a cannibalistic prey–predator model with Allee effect in prey under the presence of diffusion

In this study, we have investigated the temporal and spatio-temporal behavior of a prey–predator model with weak Allee effect in prey and the quality of being cannibalistic in a specialist predator. The parameters responsible for the Allee effect and cannibalism impact both the existence and stability of coexistence steady states of the temporal system. The temporal system exhibits various kinds of local bifurcations such as saddle–node, Hopf, Generalized Hopf (Bautin), Bogdanov–Takens, and global bifurcation like homoclinic, saddle–node bifurcation of limit cycles. For the model with self-diffusion, we establish the non-negativity and prior bounds of the solution. Subsequently, we derive the theoretical conditions in which self-diffusion leads to the destabilization of the interior equilibrium. Additionally, we explore the conditions under which cross-diffusion induces the Turing-instability where self-diffusion fails to do so. Further, we present different kinds of stationary and dynamic patterns on varying the values of diffusion coefficients to depict the spatio-temporal model’s rich dynamics. It has been found that the addition of self and cross-diffusion in a prey–predator model with the Allee effect in prey and cannibalistic predator play essential roles in comprehending the pattern formation of a distributed population model. It is expected that the comprehensive mathematical analysis and extensive numerical simulations used in investigating the global dynamics of the proposed model can facilitate researchers in studying the temporal and spatial aspects of prey–predator models in more significant detail.

Dynamics of a prey–predator model with reproductive Allee effect for prey and generalist predator

Dynamics of a prey–predator model with reproductive Allee effect for prey and generalist predator

Dynamics of a Class of Prey–Predator Models with Singular Perturbation and Distributed Delay

In this paper, two prey–predator models with distributed delays are presented based on the growth and loss rates of the predator, which are much smaller than that of the prey, leading to a singular perturbation problem. It is obtained that Hopf bifurcation can occur, where the coexistence equilibrium becomes unstable leading to a stable limit cycle. Subsequently, considering the perturbation parameter [Formula: see text], the fact that the solution crossing the transcritical point converges to a stable equilibrium is discussed for the model with Holling type I using the linear chain criterion, center-manifold reduction, the geometric singular perturbation theory and entry–exit function. The existence and uniqueness of relaxation oscillation cycle for the model with Holling type II are obtained. In addition, numerical simulations are provided to verify the analytical results.

Bioenergetic modelling of a marine top predator's responses to changes in prey structure.

Determining how animals allocate energy, and how external factors influence this allocation, is crucial to understand species' life history requirements and response to disturbance. This response is driven in part by individuals' energy balance, prey characteristics, foraging behaviour and energy required for essential functions. We developed a bioenergetic model to estimate minimum foraging success rate (FSR), that is, the lowest possible prey capture rate for individuals to obtain the minimum energy intake needed to meet daily metabolic requirements, for female sperm whale (Physeter macrocephalus). The model was based on whales' theoretical energetic requirements using foraging and prey characteristics from animal-borne tags and stomach contents, respectively. We used this model to simulate two prey structure change scenarios: (1) decrease in mean prey size, thus lower prey energy content and (2) decrease in prey size variability, reducing the variability in prey energy content. We estimate the whales need minimum of ~14% FSR to meet their energetic requirements, and energy intake is more sensitive to energy content changes than a decrease in energy variability. To estimate vulnerability to prey structure changes, we evaluated the compensation level required to meet bioenergetic demands. Considering a minimum 14% FSR, whales would need to increase energy intake by 21% (5-35%) and 49% (27-67%) to compensate for a 15% and 30% decrease in energy content, respectively. For a 30% and 50% decrease in energy variability, whales would need to increase energy intake by 13% (0-23%) and 24% (10-35%) to meet energetic demands, respectively. Our model demonstrates how foraging and prey characteristics can be used to estimate impact of changing prey structure in top predator energetics, which can help inform bottom-up effects on marine ecosystems. We showed the importance of considering different FSR in bioenergetics models, as it can have decisive implications on estimates of energy acquired and affect the conclusions about top predator's vulnerability to possible environmental fluctuations.

Discrete kinetic analysis of a general reaction–diffusion model constructed by Euler discretization and coupled map lattices

In this paper, we provide a framework for the Turing instability in a general two-dimensional discrete reaction–diffusion model that utilizes Euler discretization and coupled map lattices. We obtained explicit criterions for the normal forms of Neimark–Sacker bifurcation and flip bifurcation in the absence of diffusion. We derive the general conditions that govern the emergence of pure Turing instability, Neimark–Sacker–Turing instability, Flip–Turing instability, spatially homogeneous stable states, spatially homogeneous periodic oscillation states, and spatially homogeneous period-doubling oscillatory states under spatially inhomogeneous conditions. As an example, we employ a ratio-dependent predation model and verify the derived general conditions through a series of numerical simulations. Additionally, we apply the calculation of Maximum Lyapunov Exponent(MLE) to simulate and demonstrate the path from flip bifurcation to chaos.

Exploring differences in neophobia and anti-predator behaviour between urban and rural mountain chickadees

Abstract Urbanization is changing natural landscapes worldwide, pushing species to quickly acclimate or adapt if they are to survive in urban environments. Mountain chickadees (Poecile gambeli) readily nest in both urban and rural environments without suffering apparent reproductive costs. However, whether urban-nesting chickadees are successful in these environments due to differences in behaviour between urban and rural birds remains untested. We examined the behavioural responses of urban and rural nesting mountain chickadee females when presented with a novel object (red plastic cup) or simulated predator (imitation squirrel model) at the nest. Behavioural responses depended on both the type of model and the habitat. As expected, mountain chickadees responded more strongly to squirrel models than novel objects; however, the magnitude of the difference in response depended on habitat. Urban birds seemingly ignored the novel object, spending little time investigating, and re-entering the nest box quickly. In contrast, rural birds spent more time reacting to the novel object and alarm calling within 5 m of the nest. When presented with a predator model, the urban birds reacted relatively more strongly (compared to the novel stimulus) than rural birds, spending more time within 5 m of the nest and alarm calling. These results suggest that either mountain chickadees in urban environments quickly acclimatize to the presence of novel objects or, potentially, that less neophobic birds disproportionately settle in urban environments or experience positive selection in urban areas. Either way, reduced neophobia may aid in mountain chickadees’ ability to readily and successfully nest in such habitats.

The dynamics of an eco-epidemiological prey–predator model with infectious diseases in prey

In this paper, we first propose an eco-epidemiological prey–predator model with infectious diseases in prey. Then study the ODE model, and diffusive model with the homogeneous Neumann and Dirichlet boundary conditions, respectively. For the ODE model and the diffusive model with the homogeneous Neumann boundary conditions, we give a complete conclusion about the stabilities of nonnegative equilibrium states (nonnegative constant equilibrium solutions). The results show that these two problems has no periodic solutions, and the diffusive model with the homogeneous Neumann boundary conditions has no yet Turing patterns. For the diffusive model with the homogeneous Dirichlet boundary conditions, we first establish the necessary and sufficient conditions for the existence of positive equilibrium solutions, and prove that the positive equilibrium solution is unique when it exists. Then we study the global asymptotic stabilities of trivial and semi-trivial nonnegative equilibrium solutions.

Stabilization of Fuzzy Prey-Predator Model Using Backstepping Method

Ecological system depends on prey predator interaction and therefore, as a result, diseases may spread among prey or predator or both of them. In this work, the fuzzy logic-based systems are used to elaborate a prey-predator model to study the effect of varying in the inflection rate. Formulation of prey predator model using fuzzy logic is more realistic depiction of the phenomena, since the initial population estimates may not be precisely known in the real-life situation, therefore the initial conditions may also be considered as fuzzy. The dynamical behaviour of the fuzzy exploited system is studied by using the backstepping method. Some references working on prey-predator model, in which they used classical control schemes with a very long and complicated steps. While the proposed method of this paper simplifies the work steps. Numerical simulation results are presented to validate the theoretical analysis.

Periodic behavior and dynamical analysis of a prey–predator model incorporating the Allee effect and fear effect

Periodic behavior and dynamical analysis of a prey–predator model incorporating the Allee effect and fear effect

Analysis of a stochastic model for a prey–predator system with an indirect effect

In the current work, we develop and analyse a prey–predator model as a semi-Kolmogorov population model in which the predator has an indirect effect on the prey. The functional response of the model is investigated as Holling type (II). We construct a stochastic environment because of the parameter's random essence to study the influence of environmental fluctuations on this model and introduce a stochastic version of the prey–predator model. Then we present a dynamical analysis of solutions, including existence, uniqueness, positivity, stochastic boundedness, and stochastic extinction of all prey. Finally, some numerical simulations are carried out to validate our theoretical findings and confirm the efficiency and adaptation of our stochastic model.