Crowley Martin Functional Response Research Articles

Explosive predator and mutualistic preys: A comparative study

Mutualism is a powerful strategy by which species can assist each other to survive in various harsh environments. These will often take the form of anti-predator strategies. In the current manuscript we consider two preys that mutually benefit each other in the presence of a sexually reproductive generalist predator. In the absence of the predator, the prey grow logistically, and there is no mutualism between them. However, in the presence of the predator the prey will begin to cooperate. We formulate five different models according to their functional responses which are either prey dependent (Holling type-III, IV and Hassel Varley functional responses) or predator–prey dependent (Beddington–DeAngelis and Crowley–Martin functional responses). We perform dynamical analysis of these five models, and also establish the mathematical restrictions under which the species can blow up at a finite time. Interestingly it is observed that the prey populations will blow-up before the predator can. We use our results to comment on a certain paradox in ecological theory, as well as provide further insight into the nature of predator interference and exploding populations of invasive species.

Qualitative analysis of a diffusive Crowley–Martin predator–prey model: the role of nonlinear predator harvesting

In this paper, we discuss a diffusive predator–prey system with mutually interfering predator and nonlinear harvesting in predator with Crowley–Martin functional response. The mathematical analysis of the system starts with the existence and uniqueness of solution of the system using $$C_0$$ semigroup. The analysis reflects that the upper bound of rate of predator harvesting for the coexistence of the species can be guaranteed. In addition, we establish the existence and nonexistence of non-constant positive steady state. Explicit conditions on predator harvesting are obtained for local and global stability of interior equilibrium and also for the existence and nonexistence of non-constant steady-state solution. We also investigate conditions for Turing instabilities of the diffusive system analytically. Our results show that the effort of harvesting (g) provides a threshold value for existence of non-constant positive stationary solution. Furthermore, we illustrate the spatial patterns via numerical simulations, which show that the system exhibits interesting patterns. Some biological implications of obtained theoretical results have also been discussed.

A delayed eco-epidemiological model with nonlinear incidence rate and Crowley–Martin functional response for infected prey and predator

A prey–predator model with disease in prey population is considered here. The prey population is classified into two categories: susceptible prey and infected prey. The nonlinear incidence rate, at which the susceptible prey becomes infected, is incorporated. Here, predator consumes infected prey only according to Crowely–Martin functional response. Time lag is introduced to convert the susceptible prey to infected one, and the present model becomes delayed eco-epidemic model. The positivity, boundedness and permanence of the solutions of both models are guaranteed here. The existence of equilibrium points is assured. Local as well as global stability analyses of both the models around the equilibrium points are performed and finally summarized in a table, comparing the conditions between two models. The occurrence of bifurcation (saddle-node, transcritical, Hopf and Bogdanov–Takens) in the non-delay model is explored here. The nature and direction of Hopf bifurcation are searched using normal form and centre manifold theorems for delay model. Lastly, the numerical simulation is carried out to verify the analytical findings. An effort is made to search the parameters, by paying special attention to newly introduced parameters, which control the dynamics of both models. Accordingly, stable/unstable zones on $$x\tau _\mathrm{c}$$ -planes, where x is any considered parameter and $$\tau _\mathrm{c}$$ is the value of critical delay time, are proposed. The stranger attractor is searched in the delay model. Bifurcated diagrams of population with respect to newly introduced parameters are exemplified and discussed. When the population (S, I and P) starts oscillating, their time-averaged values are presented to show their variation due to aforesaid parameters. Moreover, pattern of all the curves $$S=S(x)$$ , $$I=I(x)$$ and $$P=P(x)$$ , where x is any aforesaid parameters, are certified using their respective gradient.

Dynamic analysis of a reaction–diffusion impulsive hybrid system

In this paper, a predator–prey system with Crowley–Martin functional response, which is described by a couple of reaction–diffusion equations with impulsive, is studied analytically and numerically. The aim of this research is to analyze how the impulsive effect influences dynamics of the system. Dynamics of the system, including the ultimate boundedness, permanence and extinction, are investigated firstly under impulsive effects. Significantly, it is found that there exists a unique positive periodic solution that is globally asymptotically stable when impulsive effects reach some critical state. Additionally, a series of numerical simulations are carried out to further study the dynamics of the system, which are consistent with the analytical results.

Extinction Analysis of Stochastic Predator-Prey System with Stage Structure and Crowley-Martin Functional Response.

In this paper, we researched some dynamical behaviors of a stochastic predator–prey system, which is considered under the combination of Crowley–Martin functional response and stage structure. First, we obtained the existence and uniqueness of the global positive solution of the system. Then, we studied the stochastically ultimate boundedness of the solution. Furthermore, we established two sufficient conditions, which are separately given to ensure the stochastic extinction of the prey and predator populations. In the end, we carried out the numerical simulations to explain some cases.

Cross Diffusion Induced Turing Patterns in a Tritrophic Food Chain Model with Crowley-Martin Functional Response

Diffusion has long been known to induce pattern formation in predator prey systems. For certain prey-predator interaction systems, self diffusion conditions ceases to induce patterns, i.e., a non-constant positive solution does not exist, as seen from the literature. We investigate the effect of cross diffusion on the pattern formation in a tritrophic food chain model. In the formulated model, the prey interacts with the mid level predator in accordance with Holling Type II functional response and the mid and top level predator interact via Crowley-Martin functional response. We prove that the stationary uniform solution of the system is stable in the presence of diffusion when cross diffusion is absent. However, this solution is unstable in the presence of both self diffusion and cross diffusion. Using a priori analysis, we show the existence of a inhomogeneous steady state. We prove that no non-constant positive solution exists in the presence of diffusion under certain conditions, i.e., no pattern formation occurs. However, pattern formation is induced by cross diffusion because of the existence of non-constant positive solution, which is proven analytically as well as numerically. We performed extensive numerical simulations to understand Turing pattern formation for different values of self and cross diffusivity coefficients of the top level predator to validate our results. We obtained a wide range of Turing patterns induced by cross diffusion in the top population, including floral, labyrinth, hot spots, pentagonal and hexagonal Turing patterns.

HOPF BIFURCATION ANALYSIS OF A DENSITY PREDATOR-PREY MODEL WITH CROWLEY-MARTIN FUNCTIONAL RESPONSE AND TWO TIME DELAYS

In this paper, a delayed density dependent predator-prey model with Crowley-Martin functional response and two time delays for the predator is considered. By analyzing the corresponding characteristic equations, the local stability of each of the feasible equilibria of the system is addressed and the existence of Hopf bifurcation at the coexistence equilibrium is established. With the help of normal form method and center manifold theorem, some explicit formulas determining the direction of Hopf bifurcation and the stability of bifurcating period solutions are derived. Finally, numerical simulations are given to illustrate the theoretical results.

STABILITY ANALYSIS OF TIME DELAYED FRACTIONAL ORDER PREDATOR-PREY SYSTEM WITH CROWLEY-MARTIN FUNCTIONAL RESPONSE

In this paper, we present a fractional order predator-prey system with Crowley-Martin functional response. Firstly, we analyze the asymptotic stability of the system. At the same time, some sufficient conditions for the stability of the system are given. Then, we investigate the stability of the corresponding system with time delay and also discuss some sufficient conditions for the equilibrium stability of the system with time delay. In the end, the numerical simulations illustrate the accuracy of our conclusions.

DYNAMICS OF A MODIFIED LESLIE-GOWER MODEL WITH GESTATION EFFECT AND NONLINEAR HARVESTIN

This study focuses on the dynamics of a modified Leslie-Gower predator-prey model where the intake rate of prey is by per capita predator according to Crowley-Martin functional response and prey is harvested through nonlinear harvesting strategy. Further the time-delay $(\tau)$ is imposed to utilize gestation period of predations. We investigate the permanence analysis of proposed system. The local stability of non-delayed model at all possible equilibrium points is studied. It is shown that the given model undergoes Hopf bifurcation around positive equilibrium point with respect to delay parameter $\tau$. Subsequently the stability of Hopf bifurcation and its direction are explored through normal and center manifold theories. The derived theoretical results are justified with the help of numerical simulations.

Interaction between prey and mutually interfering predator in prey reserve habitat: Pattern formation and the Turing–Hopf bifurcation

In this paper, we propose a diffusive prey-predator system with mutually interfering predator (Crowley–Martin functional response) and prey reserve. In particular, we develop and analyze both spatially homogeneous model based on ordinary differential equations and reaction-diffusion model. We mainly investigate the global existence and boundedness of positive solution, stability properties of homogeneous steady state, non-existence of non-constant positive steady state, conditions for Turing instability and Hopf bifurcation of the diffusive system analytically. Conventional stability properties of the non-spatial counterpart of the system are also studied. The analysis ensures that the prey reserve leaves stabilizing effect on the stability of temporal system. The prey reserve and diffusive parameters leave parallel impact on the stability of the spatio-temporal system. Furthermore, we illustrate the spatial patterns via numerical simulations, which show that the model dynamics exhibits diffusion controlled pattern formation by different interesting patterns.

Asymptotic stability of one prey and two predators model with two functional responses

We formulate a mathematical model to study the complex dynamical behavior of a three dimensional model consisting of one prey and two predators involving Beddington–DeAngelis and Crowley–Martin functional responses. The existence and stability conditions of the equilibrium points are analyzed. The global asymptotic stability of the interior equilibrium point, if exists, is proved by considering Lyapunov function. Several numerical simulations are performed to illustrate the theoretical analysis. The multiple states of stability are observed in one example whereas another example exhibits the global stability of interior equilibrium point.

Turing–Hopf bifurcation and spatiotemporal patterns in a diffusive predator–prey system with Crowley–Martin functional response

A diffusive predator–prey system with Crowley–Martin functional response is considered. Firstly, the maximal parameter region, where the coexistence equilibrium is stable, is provided, of which the boundary consists of Turing bifurcation curves and Hopf bifurcation curve, and result derived by Shi and Ruan (2015) is improved. Meanwhile, critical conditions for Turing instability are derived, which are necessary and sufficient. Furthermore, considering the degenerated situation where Turing bifurcation and Hopf bifurcation occur simultaneously, conditions for codimension-two Turing–Hopf bifurcation and Turing–Turing bifurcations are given. For Turing–Hopf bifurcation, by analyzing the normal forms truncated to order 3, which are derived by applying normal form method and generic formulas developed by Jiang, An and Shi (2018), it is found that system exhibits spatial, temporal and spatiotemporal patterns, like transient spatially inhomogeneous periodic solutions, as well as tristable phenomena of a pair of spatially inhomogeneous steady states and a spatially homogeneous periodic solution coexisting. At last, numerical simulations, including transient, bistable and tristable patterns, are illustrated to support our theory results.

Size-dependent predation and intraspecific inhibition of an estuarine snail feeding on oysters

Predator outbreaks have increased in the past two decades in many ecosystems and are predicted to become more common with climate change. During these outbreaks, predator densities increase rapidly, and can cause large reductions in prey populations or shifts in prey size structure. However, unexpected interactions may occur at high predator densities, necessitating a mechanistic understanding of how increased predator density affects predator-prey dynamics. In the northern Gulf of Mexico, outbreaks of southern oyster drill, Stramonita haemastoma (Linnaeus, 1767), occur during high salinity events, and can greatly reduce eastern oyster, Crassostrea virginica (Gmelin, 1791), populations. A large outbreak of drills occurred from 2013 to 2015 in Apalachicola Bay, FL which corresponded with an oyster fishery collapse in that bay. To improve our mechanistic understanding of predation during such a high-density outbreak, laboratory experiments based on field observations were used to quantify the prey size selection by drills and the drill functional response, as a function of drill abundance. Drills fed on medium-sized oysters (50–75 mm) more often than small and larger-sized oysters, and often formed aggregations during feeding events. However, despite this aggregative response, there was a negative relationship between per capita feeding rates and drill abundance. Indeed, the Crowley-Martin functional response model had the most parsimonious fit to the data, suggesting that predator-predator inhibition reduced attack rates and increased handling times. Due to an increase in regional drought conditions and water usage in the southeastern United States, drill outbreaks will likely increase in frequency and duration. A greater understanding of how predation rates change with predator densities during outbreaks will improve predictions of oyster mortality, and strengthen the scientific framework for oyster fishery decisions.

DYNAMIC RELATIONSHIP BETWEEN THE MUTUAL INTERFERENCE AND GESTATION DELAYS OF A HYBRID TRITROPHIC FOOD CHAIN MODEL

We have proposed a three-species hybrid food chain model with multiple time delays. The interaction between the prey and the middle predator follows Holling type (HT) II functional response, while the interaction between the top predator and its only food, the middle predator, is taken as a general functional response with the mutual interference schemes, such as Crowley–Martin (CM), Beddington–DeAngelis (BD) and Hassell–Varley (HV) functional responses. We analyse the model system which employs HT II and CM functional responses, and discuss the local and global stability analyses of the coexisting equilibrium solution. The effect of gestation delay on both the middle and top predator has been studied. The dynamics of model systems are affected by both factors: gestation delay and the form of functional responses considered. The theoretical results are supported by appropriate numerical simulations, and bifurcation diagrams are obtained for biologically feasible parameter values. It is interesting from the application point of view to show how an individual delay changes the dynamics of the model system depending on the form of functional response.

Stability and Bifurcation of a Fishery Model with Crowley–Martin Functional Response

To understand the dynamics of a fishery system, a nonlinear mathematical model is proposed and analyzed. In an aquatic environment, we considered two populations: one is prey and another is predator. Here both the fish populations grow logistically and interaction between them is of Crowley–Martin type functional response. It is assumed that both the populations are harvested and the harvesting effort is assumed to be dynamical variable and tax is considered as a control variable. The existence of equilibrium points and their local stability are examined. The existence of Hopf-bifurcation, stability and direction of Hopf-bifurcation are also analyzed with the help of Center Manifold theorem and normal form theory. The global stability behavior of the positive equilibrium point is also discussed. In order to find the value of optimal tax, the optimal harvesting policy is used. To verify our analytical findings, an extensive numerical simulation is carried out for this model system.

Stationary patterns of a diffusive predator–prey model with Crowley–Martin functional response

A diffusive predator–prey system with predator interference and Neumann boundary conditions is considered in this paper. We derive some results on the existence and nonexistence of nonconstant stationary solutions. It is shown that there exist no nonconstant stationary solutions when the effect of the predator interference is strong or the conversion rate of the predator is large, and nonconstant stationary solutions emerge when the diffusion rate of the predator is large.

Bifurcations and chaos in a discrete predator–prey model with Crowley–Martin functional response

In this paper, a discrete-time predator–prey model with Crowley–Martin functional response is investigated based on the center manifold theorem and bifurcation theory. It is shown that the system undergoes flip bifurcation and Neimark–Sacker bifurcation. An explicit approximate expression of the invariant curve, caused by Neimark–Sacker bifurcation, is given. The fractal dimension of a strange attractor and Feigenbaum’s constant of the model are calculated. Moreover, numerical simulations using AUTO and MATLAB are presented to support theoretical results, such as a cascade of period doubling with period-2, 4, 6, 8, 16, 32 orbits, period-10, 20, 19, 38 orbits, invariant curves, codimension-2 bifurcation and chaotic attractor. Chaos in the sense of Marotto is also proved by both analytical and numerical methods. Analyses are displayed to illustrate the effect of magnitude of interference among predators on dynamic behaviors of this model. Further the chaotic orbit is controlled to be a fixed point by using feedback control method.

Do prey handling predators really matter: Subtle effects of a Crowley-Martin functional response

Predator interference, or a decline in the per predator consumption rate as predator density increases, is generally considered a stabilizing mechanism in two-species predator-prey models. There is significant debate, as to whether prey handling predators, might interfere in the hunting process of prey searching predators, or whether these are mutually exclusive events. In the current manuscript, a three species food chain model, with strong top predator interference is considered. We prove that in terms of explosive instability/finite time blow up, sufficient interference by prey handling predators always tends to destabilize the system. The dynamics of a time delayed version, as well as the spatially explicit model are also explored. We use our results to comment on a certain paradox in ecological theory, as well as provide further insight into the nature of predator interference, and exploding populations of invasive species.

Global Dynamics for an HIV Infection Model with Crowley-Martin Functional Response and Two Distributed Delays

In this paper, an HIV dynamics model with two distributed intracellular delays incorporating Crowley-Martin functional response infection rate is investigated. The authors take into account multiple stage disease transmission and the latently infected cells (not yet producing virus) in our system. The authors consider nonnegativity, boundedness of solutions, and global asymptotic stability of the system. By constructing suitable Lyapunov functionals and using the Lyapunov-LaSalle invariance principle, the authors prove the global stability of the infected (endemic) equilibrium and the diseasefree equilibrium for time delays. The authors have proven that if the basic reproduction number R0 is less than unity, then the disease-free equilibrium is globally asymptotically stable, and if R0 is greater than unity, then the infected equilibrium is globally asymptotically stable. The results obtained show that the global dynamic behaviors of the model are completely determined by the basic reproduction number R0 and that the time delay does not affect the global asymptotic properties of the model.

A modified Leslie–Gower predator-prey interaction model and parameter identifiability

In this work, bifurcation and a systematic approach for estimation of identifiable parameters of a modified Leslie–Gower predator-prey system with Crowley–Martin functional response and prey refuge is discussed. Global asymptotic stability is discussed by applying fluctuation lemma. The system undergoes into Hopf bifurcation with respect to parameters intrinsic growth rate of predators (s) and prey reserve (m). The stability of Hopf bifurcation is also discussed by calculating Lyapunov number. The sensitivity analysis of the considered model system with respect to all variables is performed which also supports our theoretical study. To estimate the unknown parameter from the data, an optimization procedure (pseudo-random search algorithm) is adopted. System responses and phase plots for estimated parameters are also compared with true noise free data. It is found that the system dynamics with true set of parametric values is similar to the estimated parametric values. Numerical simulations are presented to substantiate the analytical findings.