Crowley Martin Functional Response Research Articles

Bistability in a predator–prey model characterized by the Crowley–Martin functional response: Effects of fear, hunting cooperation, additional foods and nonlinear harvesting

This study focuses on unraveling key factors influencing predator–prey interactions with Crowley–Martin functional response. Specifically, it explores the roles of additional food sources, harvesting practices, hunting cooperation, fear and its carry-over effects. We analyze equilibrium points and their stability properties through rigorous mathematical methods. Numerical illustrations showcase a diverse range of bifurcations including Hopf, saddle–node, and transcritical, providing a comprehensive understanding of the system’s dynamics. We find that the collaboration among predators during hunting induces instability in the system, leading to the emergence of population cycles from a stable state. Further, we place emphasis on investigating the impact of seasonal forcing by introducing time-varying parameters into our model. We reveal the emergence of periodic solutions, higher periodic solutions and chaotic dynamics due to the seasonal variations of the prey’s birth rate and the degree of hunting cooperation. We also emphasize the significance of incorporating different periodicity of seasonally forced parameters, leading to a more precise understanding of predator–prey dynamics.

A nonlinear fractional fishery resource system model with Crowley–Martin functional response under Mittag-Leffler kernel

This article studies the action of predators and the predator-dependent functional response in the fishery resource model. We employ the Atangana–Baleanu–Caputo fractional derivative to study the proposed fractional fishery resource model in the presence of predators with Crowley–Martin functional response. We present a theoretical and numerical analysis of the governing nonlinear differential equations of the model consisting of the biomass density of the fish population inside the unrestricted fishing zone, the biomass density of the fish population inside the reserve or restricted fishing zone, and the predator population. Using the fixed point theory and nonlinear analysis, we establish the existence and uniqueness results of the proposed ABC fractional fishery resource model. We establish stability analysis of the fishery model using the Ulam–Hyers stability approach. The numerical scheme of the fractional Adams–Bashforth method is provided and the approximate solutions for the model under consideration are given and discussed. We observe that an increase in the fish capturing rates increases the size of the predator population and reduces the fish subpopulations. To maintain a high number of fish species, we recommend a control measure to reduce the fish capturing rate by the predators.

Spatiotemporal flow-induced instability of predator-prey model with Crowley-Martin functional response and prey harvesting.

Ecological systems can generate striking large-scale spatial patterns through local interactions and migration. In the presence of diffusion and advection, this work examines the formation of flow-induced patterns in a predator-prey system with a Crowley-Martin functional response and prey harvesting, where the advection reflects the unidirectional flow of each species migration (or flow). Primarily, the impact of diffusion and advection rates on the stability and the associated Turing and flow-induced patterns are investigated. The theoretical implication of flow-induced instability caused by population migration, mainly the relative migrations between prey and predator, is examined, and it also shows that Turing instability is the particular condition of flow-induced instability. The influence of the relative flow of both species and prey-harvesting effort on the emerging pattern is reported. Advection impacts a wide range of spatiotemporal patterns, including bands, spots, and a mixture of bands and spots in both harvested and unharvested dynamics. We also observe the diagonally bend-type banded patterns and straight-type banded patterns due to positive and negative relative flows, respectively. Here, the increasing relative flow increases the band length. The growing harvesting effort also decreases the band length, producing a thin band and a mixture of spots and bands due to the negative and positive relative flows, respectively. One exciting result observed here is that harvesting effort drives the flow-Turing and flow-Turing-Hopf instability into pure-flow instability.

Comprehensive analysis of disease dynamics using nonlinear fractional order SEIRS model with Crowley–Martin functional response and saturated treatment

This paper presents a comprehensive study of disease spreading dynamics through the application of a nonlinear fractional order epidemic SEIRS model. By incorporating the Crowley–Martin type functional response and a saturated treatment function, the model effectively captures the intricacies of real-world epidemics. Our research establishes the existence, uniqueness, non-negativity and boundedness of the solution, while also investigating the model’s fundamental reproduction number. Additionally, we conduct a thorough analysis of the specific conditions governing the local and global stability of the model’s equilibriums. A notable observation is the variation of the reproduction number with the fractional-order [Formula: see text], which represents a memory effect on individuals’ dynamic behavior and reveals the influence of interactions between compartments. To validate these theoretical findings, we employ numerical simulations using Matlab, demonstrating that inhibition measures for susceptibles and the saturated treatment parameters play a pivotal role in determining the disease state. Specifically, we observe that as these parameter values increase, the transition from endemic equilibrium to disease-free equilibrium occurs.

Bifurcation and Turing Pattern Formation in a Diffusion Modified Leslie-Gower Predator-Prey Model with Crowley-Martin Functional Response

Bifurcation and Turing Pattern Formation in a Diffusion Modified Leslie-Gower Predator-Prey Model with Crowley-Martin Functional Response

Spatiotemporal Dynamics of a Diffusive Immunosuppressive Infection Model with Nonlocal Competition and Crowley–Martin Functional Response

In this paper, we introduce the Crowley–Martin functional response and nonlocal competition into a reaction–diffusion immunosuppressive infection model. First, we analyze the existence and stability of the positive constant steady states of the systems with nonlocal competition and local competition, respectively. Second, we deduce the conditions for the occurrence of Turing, Hopf, and Turing–Hopf bifurcations of the system with nonlocal competition, as well as the conditions for the occurrence of Hopf bifurcations of the system with local competition. Furthermore, we employ the multiple time scales method to derive the normal forms of the Hopf bifurcations reduced on the center manifold for both systems. Finally, we conduct numerical simulations for both systems under the same parameter settings, compare the impact of nonlocal competition, and find that the nonlocal term can induce spatially inhomogeneous stable periodic solutions. We also provide corresponding biological explanations for the simulation results.

Global Dynamics of a Within-Host Model for Usutu Virus

We propose a within-host mathematical model for the dynamics of Usutu virus infection, incorporating Crowley–Martin functional response. The basic reproduction number R0 is found by applying the next-generation matrix approach. Depending on this threshold, parameter, global asymptotic stability of one of the two possible equilibria is also established via constructing appropriate Lyapunov functions and using LaSalle’s invariance principle. We present numerical simulations to illustrate the results and a sensitivity analysis of R0 was also completed. Finally, we fit the model to actual data on Usutu virus titers. Our study provides new insights into the dynamics of Usutu virus infection.

Dynamics of insect predator and mosquito prey system with mutual interference as a factor for the co-occurrence: Validating through models

Several models have been proposed as an extension to the classical Holling’s disc equation to evaluate the predator and prey interactions and their applied aspects in biological control and population regulation of the target organisms. In a one-prey and two-predator dynamic system with mutual interference m as a quadratic parameter of predator density, an evaluation was made of the resultant impact on the prey. A simulation was carried out to see the finite-time extinction of prey and the stability of the system at origin, i.e., when all three species are extinct. We assumed the data obtained was for the interactions between the mosquito and the water bug predators that are common in the freshwater wetlands and involved in population regulation. Despite the benefits to the prey population due to interference and competition, the expected extinction of prey in a finite time is still observed. With varying magnitudes of m, the declining growth curve of the prey population shifted. The equation proposed was also compared with the Crowley-Martin functional response, and considerable differences were observed in selected instances when compared to the growth rate of the predators in a species-specific manner. The stability of the system was deduced from the eigenvalues of the Jacobian matrix at the origin to prove the extinction is stable. Our assessment supports the possible cooccurrence of predators and mosquito prey in the wetlands, with mutual interference being one of the major factors.

Bistability and tristability in a deterministic prey–predator model: Transitions and emergent patterns in its stochastic counterpart

The paper focuses on studying the global dynamics of a simple prey–predator model. The model incorporates cooperative hunting behavior among predators and takes into account the effect of harvesting on the predator population. The interaction between the prey and predator follows the Crowley–Martin functional response. The study involves analyzing the equilibrium points of the system and investigating their stability properties through mathematical analysis. Various types of bifurcations, including Hopf bifurcation, saddle–node bifurcation, and transcritical bifurcation, are numerically demonstrated in the figures, highlighting the dynamic behavior of the model. One of the intriguing findings of the study is the occurrence of bistability and tristability in the model. To incorporate stochasticity into the model, white noise is added to the deterministic system. This allows for the examination of transitions between different steady states in the stochastic system. We have conducted an analysis of species persistence and extinction in relation to the presence of noise. The paper presents the stochastic sensitivity function (SSF) technique and the use of confidence ellipses to assess the likelihood of such transitions occurring in the system. Overall, the study provides insights into the complex dynamics of prey–predator interactions, considering factors such as cooperation, harvesting, and stochasticity. The results contribute to our understanding of population dynamics and the potential effects of environmental and human-induced perturbations on ecosystem stability.

Qualitative analysis of a prey–predator model with prey refuge and intraspecific competition among predators

In this study, we consider a prey–predator model with prey refuge and intraspecific competition between predators using the Crowley–Martin functional response and investigate the dynamic characteristics of spatial and nonspatial prey–predator systems via both analytical and numerical methods. The local stability of nontrivial interior equilibrium, the existence of a Hopf bifurcation, and the stability of bifurcating periodic solutions are obtained in the absence of diffusion. For the spatial system, the Turing and non-Turing patterns are evaluated for some set of parametric belief functions, and we obtain some interesting results in terms of prey and predator inhabitants. We present the results of numerical simulations that demonstrate that both prey and predator populations do not converge to a stationary equilibrium state at any foreseeable future time when the parametric values are processed in the Turing domain.

Existence of positive periodic solutions for a periodic predator–prey model with fear effect and general functional responses

This paper investigates the existence of positive periodic solutions for a periodic predator-prey model with fear effect and general functional responses. The general functional responses can cover the Holling types II and III functional response, the Beddington–DeAngelis functional response, the Crowley–Martin functional response, the ratio-dependent type with Michaelis–Menten type functional response, etc. Some new sufficient conditions for the existence of positive periodic solutions of the model are obtained by employing the continuation theorem of coincidence degree theory and some ingenious estimation techniques for the upper and lower bounds of the a priori solutions of the corresponding operator equation. Our results considerably improve and extend some known results.

Dynamics of a Coccinellids-Aphids Model with Stage Structure in Predator Including Maturation and Gestation Delays

This work studies a three-dimensional predator–prey model with gestation delay and stage structure between aphidophagous coccinellids and aphid pests, where the interaction between mature coccinellids and aphids is inscribed by Crowley–Martin functional response function, and immature coccinellids and aphids act in the form of Holling-I type. We prove the positivity and boundedness of the solution of the nondelayed system and analyze its equilibrium point, local asymptotic stability, and global stability. In addition to the delays, the critical values of Hopf bifurcation occurring for different parameters are also found from the numerical simulation. The stability of the delayed system and Hopf bifurcation with different delays as parameters are also discussed. Our model analysis shows that the time delay essentially governs the system’s dynamics, and the stability of the system switches as delays increase. We also investigate the direction and stability of the Hopf bifurcation using the normal form theory and center manifold theorem. Finally, we perform computer simulations and depict diagrams to support our theoretical results.

Hopf Bifurcation for an Age-Structured Predator–Prey Model with Crowley–Martin Functional Response and Two Delays

Hopf Bifurcation for an Age-Structured Predator–Prey Model with Crowley–Martin Functional Response and Two Delays

A stage-structured predator-prey model with Crowley-Martin functional response

<p style='text-indent:20px;'>A stage-structured predator-prey model with Crowley-Martin functional response is formulated and analyzed to investigate the impaction of predator maturity delay and predator interference on the dynamics of the system. It is shown that the net reproduction number of the system determines its threshold dynamics: the system is permanent if and only if the net reproduction number is larger than one unit, otherwise the predators go extinct; Moreover, given that the system is permanent, the unique coexistence equilibrium is shown to be globally asymptotically stable provided that the predator's interference is large enough; Numerical simulations are also performed to explore the effects of predator maturity delay and predator's interference on the stability of interior equilibrium. It is indicated that an increase of maturity delay of the predator will enlarge its extinction risk and may also enrich dynamics of the system in the sense of bringing stability switches of the coexistence equilibrium; and that an increase of predator's interference may lead to stability switch of coexistence equilibrium from unstable to stable, implying that a large predator's interference can be stabilizing.</p>

The effects of fear and delay on a predator-prey model with Crowley-Martin functional response and stage structure for predator

<abstract><p>Taking into account the delayed fear induced by predators on the birth rate of prey, the counter-predation sensitiveness of prey, and the direct consumption by predators with stage structure and interference impacts, we proposed a prey-predator model with fear, Crowley-Martin functional response, stage structure and time delays. By use of the functional differential equation theory and Sotomayor's bifurcation theorem, we established some criteria of the local asymptotical stability and bifurcations of the system equilibrium points. Numerically, we validated the theoretical findings and explored the effects of fear, counter-predation sensitivity, direct predation rate and the transversion rate of the immature predator. We found that the functional response as well as the stage structure of predators affected the system stability. The fear and anti-predation sensitivity have positive and negative impacts to the system stability. Low fear level and high anti-predation sensitivity are beneficial to the system stability and the survival of prey. Meanwhile, low anti-predation sensitivity can make the system jump from one equilibrium point to another or make it oscillate between stability and instability frequently, leading to such phenomena as the bubble, or bistability. The fear and mature delays can make the system change from unstable to stable and cause chaos if they are too large. Finally, some ecological suggestions were given to overcome the negative effect induced by fear on the system stability.</p></abstract>

Dynamical Behaviours of Stage-Structured Fractional-Order Prey-Predator Model with Crowley–Martin Functional Response

In this paper, the dynamic behaviour of the stage-structure prey-predator fractional-order derivative system is considered and discussed. In this model, the Crowley–Martin functional response describes the interaction between mature preys with a predator. The existence, uniqueness, non-negativity, and the boundedness of solutions are proved. All possible equilibrium points of this system are investigated. The sufficient conditions of local stability of equilibrium points for the considered system are determined. Finally, numerical simulation results are carried out to confirm the theoretical results.

Global stability of a general HTLV-I infection model with Cytotoxic T-Lymphocyte immune response and mitotic transmission

Mathematical models have been considered as a robust tool to support biological and medical studies of human viral infections. The global stability of viral infection models remains an important and largely open research problem. Such results are necessary to evaluate treatment strategies for infections and to establish thresholds for treatment rates. Human T-lymphotropic virus class I (HTLV-I) is a retrovirus which infects the CD4+T cells and causes chronic and deadly diseases. In this article, we developed a general nonlinear system of ODEs which describes the within-host dynamics of HTLV-I under the effect Cytotoxic T-Lymphocytes (CTLs) immunity. The mitotic division of actively infected cells are modeled. We consider general nonlinear functions for the generation, proliferation and clearancerates for all types of cells. The incidence rate of infection is also modeled by a general nonlinear function. These general functions are assumed to satisfy a set of suitable conditions and include several forms presented in the literature. We determine a bounded domain for the system’s solutions. We prove the existence of the system’s equilibrium points and determine two threshold numbers, the basic reproductive number R0 and the CTL immunity stimulation number R1. We establish the global stability of all equilibrium points by constructing Lyapunov function and applying Lyapunov-LaSalle asymptotic stability theorem. Under certain conditions it is shown that ifR0≤1, then the infection-free equilibrium point is globally asymptotically stable (GAS) and the HTLV-I infection is cleared. If R1≤1<R0, then the infected equilibrium point without CTL immunity is GAS and the HTLV-I infection becomes chronic with no sustained CTL immune response. If R1>1, then the infected equilibrium point with CTL immunity is GAS and the infection becomes chronic with persistent CTL immune response. We present numerical simulations for the system by choosing special shapes of the general functions. The effect of Crowley-Martin functional response and mitotic division of actively infected cells on the HTLV-I progression are studied. Our results cover and improve several ones presented in the literature.

Complex dynamical behaviour of predator–prey model with harvesting and multiple delays

Abstract In this work, we investigate a predator–prey model with Crowley–Martin functional response and constant harvesting. The model is extended by incorporating two constant time delays, where the first delay(τ 1) is for density dependent feedback mechanism in the logistic growth of the prey and the second one is for gestation delay(τ 2) of the predator population. The dynamical behaviours such as positivity, boundedness, extinction criteria and existence, stability and bifurcations of the equilibria of the non-delay model are qualitatively discussed. The existence of periodic solutions via Hopf-bifurcation with respect to absence of delay, single delay and both delays are established. Finally, numerical simulations have been carried out to confirm our numerical results.

Dynamic Behaviors of an Obligate Commensal Symbiosis Model with Crowley–Martin Functional Responses

A two species obligate commensal symbiosis model with Crowley–Martin functional response was proposed and studied in this paper. For an autonomous case, local and global dynamic behaviors of the system were investigated, respectively. The conditions that ensure the existence of the positive equilibrium is coincidentla to the conditions of global stability of a positive equilibrium. For nonautonomous cases, persistent and extinction properties of the system are investigated.

DYNAMIC BEHAVIOR OF A STOCHASTIC NON-AUTONOMOUS PREDATOR–PREY MODEL WITH CROWLEY–MARTIN FUNCTIONAL RESPONSE AND IMPULSES

A stochastic non-autonomous one-prey two-predator model with Crowley–Martin functional response and impulses is proposed in this paper. First, by constructing the equivalent system without impulses, we investigate the existence and uniqueness of the global positive solution of the system. Second, by using Itô formula, strong law of large numbers and Chebyshev’s inequality, some sufficient conditions are established to ensure the extinction, non-persistence in the mean, persistence in the mean and stochastic permanence of the system. Third, we prove the system is globally attractive under some conditions. Finally, we choose different white noise intensities and impulsive parameters to illustrate the analytical results by numerical simulations.