A delay digestion process with application in a three-species ecosystem

In a natural ecosystem, specialist predators feed almost exclusively on one specific species of prey which may be possible for a parasitoid. But generalist predators feed on many types of species. It is also well known that the predation rate increases as prey density rises, but eventually levels off due to the predator’s handling time. The response function, thereby, is often assumed to Holling II functional response. In addition, digestion processes of the predation often involve reactions with delays. In view of these facts, a three-species ecosystem with a delay digestion process and Holling functional response is formulated. By analyzing the corresponding characteristic equations, the stability of the equilibria is investigated. Furthermore, Hopf bifurcations occurring at the positive equilibrium under some conditions are demonstrated. The consequence of global stability of the positive equilibrium is that predation will not irreversibly change the system. That is, as long as preys are not made extinct by excessive predation of their predator, the system is able to recover. Numerical simulations are carried out to illustrate our theoretical results. Meanwhile, they indicate that time delay is harmless for permanence of populations even thought it has a tendency to produce oscillations.     

A Predator�CPrey Model with a Holling Type I Functional Response Including a Predator Mutual Interference

The most widely used functional response in describing predator–prey relationships is the Holling type II functional response, where per capita predation is a smooth, increasing, and saturating function of prey density. Beddington and DeAngelis modified the Holling type II response to include interference of predators that increases with predator density. Here we introduce a predator-interference term into a Holling type I functional response. We explain the ecological rationale for the response and note that the phase plane configuration of the predator and prey isoclines differs greatly from that of the Beddington–DeAngelis response; for example, in having three possible interior equilibria rather than one. In fact, this new functional response seems to be quite unique. We used analytical and numerical methods to show that the resulting system shows a much richer dynamical behavior than the Beddington–DeAngelis response, or other typically used functional responses. For example, cyclic-fold, saddle-fold, homoclinic saddle connection, and multiple crossing bifurcations can all occur. We then use a smooth approximation to the Holling type I functional response with predator mutual interference to show that these dynamical properties do not result from the lack of smoothness, but rather from subtle differences in the functional responses.     

An eco‐epidemiological predator–prey model where predators distinguish between susceptible and infected prey

A predator–prey model with disease amongst the prey and ratio‐dependent functional response for both infected and susceptible prey is proposed and its features analysed. This work is based on previous mathematical models to analyse the important ecosystem of the Salton Sea in Southern California and New Mexico where birds (particularly pelicans) prey on fish (particularly tilapia). The dynamics of the system around each of the ecologically meaningful equilibria are presented. Natural disease control is considered before studying the impact of the disease in the absence of predators and the interaction of predators and healthy prey and the disease effects on predators in the absence of healthy prey. Our theoretical results are confirmed by numerical simulation. Copyright © 2016 John Wiley & Sons, Ltd.     

A predator-prey system with L data

In this paper, we are concerned with a system of nonlinear partial differential equations modeling a predator-prey system in heterogeneous habitats. Preys are assumed to follow a logistic growth in the absence of predation, and predators are assumed to feed on preys with a Holling type II functional response to prey density. Also, interactions between predators are modelized by the statement of a food pyramid condition. Assuming no-flux boundary conditions and L¹ data, we prove the existence of at least one weak solution.     

Global Analysis of Predator–Prey System with Hawk and Dove Tactics

Functional response of the Holling type II is incorporated into a predator–prey model with predators using hawk‐dove tactics to consider combination effects of nonlinear functional response and individual tactics. By mathematical analysis, it is shown that the model undergoes a sequence of bifurcations including saddle‐node bifurcation, supercritical Hopf bifurcation and homoclinic bifurcation. New phenomena are found that include the bistable coexistence of prey and predators in the form of a stable limit cycle and a stable positive equilibrium, the bistable coexistence of prey and predators in a large stable limit cycle that encloses three positive equilibria and a stable positive equilibrium within the cycle, and the bistable coexistence of two stable limit cycles.     

Alternative prey source coupled with prey recovery enhance stability between migratory prey and their predator in the presence of disease

An eco-epidemiological model is considered where the prey population is migratory in nature. To incorporate the temporal pattern of the avian migration into the model, a time dependent recruitment rate was considered with a general functional response. In the numerical simulation we substitute the general functional response with Holling type-I and Holling type-II functional responses. It was observed that the qualitative behaviour of the system does not depend on the choice of the functional responses. The results showed that the system could be made disease free by either decreasing the contact rate or simultaneously increasing the predation and the recovery rate. Moreover, it was observed that the presence of an alternative food source for the predator population helps in the coexistence of all the species.     

Stability and bifurcation analysis of a prey–predator model with age based predation

It is observed that in large animals only adult predators take part in direct predation while suckling feed on milk of adult predators and juveniles are dependent on the dead prey stock killed by the adult predators. Some parts of the dead prey population is consumed by adult predators and remaining parts are consumed by juveniles and the remaining portion decays naturally. In light of this, a mathematical model is proposed to study the stability and bifurcation behaviour of a prey–predator system with age based predation. All the feasible equilibria of the system are obtained and the conditions for the existence of the interior equilibrium are determined. The local stability analysis of all the feasible equilibria is carried out and the possibility of Hopf-bifurcation of the interior equilibrium is studied. Finally, numerical simulation is conducted to support the analytical results.     

Neimark–Sacker bifurcation analysis in an intraguild predation model with general functional responses

We analyse the dynamics of a discrete system coming from an intraguild food web model by using the average method. The intraguild predation model is formed by three populations corresponding to prey (P), mesopredator (MP) and superpredator (SP), where these last two populations are specialist. We give sufficient condition to guarantee the existence of a coexistence point at which the intraguild predation discrete model undergoes a Neimark–Sacker bifurcation independently of the functional responses that govern the interactions. We show numerical applications that consist in to assume that P has logistic growth and that the relation of MP–P is through a Holling type II functional response. Besides, we will consider that the interaction of MP–P is such that population MP has defense. The interaction of SP–P will be through a Holling functional response type III or IV. In particular, we give sufficient conditions to guarantee that the three species coexist. The techniques used to obtain the results can be applied to other models with different functional responses.     

Ecoepidemic models with disease incubation and selective hunting

In this paper we consider ecoepidemic models in which the basic demographics is represented by predator-prey interactions, with the disease modeled by an SEI system. At first we consider a basic Lotka-Volterra type of interaction. Then we also introduce competition for resources among individuals of the prey population. Several variations of the model are presented, in which the prey intra-specific population pressure assumes different forms, depending on the virulence of the disease. Indeed, the latter may affect the exposed and infected individuals so much that they may not be able to compete with the sound ones for resources. A further distinguishing feature of this investigation lies in the way in which the predator actively selects the prey for hunting. For instance in some cases predators may discard the diseased ones, as less palatable, while in other situations they would instead search expressly for the infected, since these are weaker individuals and thus easier to hunt. The equilibria of the systems are analyzed, showing that in some cases bifurcations arise, contrary to what happens to similar classical Holling type I ecoepidemic models. These persistent oscillations seem to be triggered by the number of subpopulations present in the system, which is larger than those introduced in the former models, counting also the latent class. Furthermore, adding predation to an SEI epidemic model has profound effects on the stability of its equilibria. In particular, once the predators are introduced into an SEI epidemic at a stable endemic equilibrium, their presence destabilizes this equilibrium making the previous stable conditions unrecoverable.     

A MICRO‐LEVEL ‘CONSUMER APPROACH’ TO SPECIES POPULATION DYNAMICS

ABSTRACT. In this paper we develop a micro ecosystem model whose basic entities are representative organisms which behave as if maximizing their net offspring under constraints. Net offspring is increasing in prey biomass intake, declining in the loss of own biomass to predators and Allee's law applies. The organism's constraint reflects its perception of how scarce its own biomass and the biomass of its prey is. In the short‐run periods prices (scarcity indicators) coordinate and determine all biomass transactions and net offspring which directly translates into population growth functions. We are able to explicitly determine these growth functions for a simple food web when specific parametric net offspring functions are chosen in the micro‐level ecosystem model. For the case of a single species our model is shown to yield the well‐known Verhulst‐Pearl logistic growth function. With two species in predator‐prey relationship, we derive differential equations whose dynamics are completely characterized and turn out to be similar to the predator‐prey model with Michaelis‐Menten type functional response. With two species competing for a single resource we find that coexistence is a knife‐edge feature confirming Tschirhart's [2002] result in a different but related model.     

General functional response and recruitment in a predator–prey system with capture on both species

This work provides a mathematical model for a predator‐prey system with general functional response and recruitment, which also includes capture on both species, and analyzes its qualitative dynamics. The model is formulated considering a population growth based on a general form of recruitment and predator functional response, as well as the capture on both prey and predators at a rate proportional to their populations. In this sense, it is proved that the dynamics and bifurcations are determined by a two‐dimensional threshold parameter. Finally, numerical simulations are performed using some ecological observations on two real species, which validate the theoretical results obtained. Copyright © 2014 John Wiley & Sons, Ltd.     

A modified Leslie–Gower predator–prey model with ratio‐dependent functional response and alternative food for the predator

In this work, a modified Leslie–Gower predator–prey model is analyzed, considering an alternative food for the predator and a ratio‐dependent functional response to express the species interaction. The system is well defined in the entire first quadrant except at the origin ( 0 , 0 ) . Given the importance of the origin ( 0 , 0 ) as it represents the extinction of both populations, it is convenient to provide a continuous extension of the system to the origin. By changing variables and a time rescaling, we obtain a polynomial differential equations system, which is topologically equivalent to the original one, obtaining that the non‐hyperbolic equilibrium point ( 0 , 0 ) in the new system is a repellor for all parameter values. Therefore, our novel model presents a remarkable difference with other models using ratio‐dependent functional response. We establish conditions on the parameter values for the existence of up to two positive equilibrium points; when this happen, one of them is always a hyperbolic saddle point, and the other can be either an attractor or a repellor surrounded by at least one limit cycle. We also show the existence of a separatrix curve dividing the behavior of the trajectories in the phase plane. Moreover, we establish parameter sets for which a homoclinic curve exits, and we show the existence of saddle‐node bifurcation, Hopf bifurcation, Bogdanov–Takens bifurcation, and homoclinic bifurcation. An important feature in this model is that the prey population can go to extinction; meanwhile, population of predators can survive because of the consumption of alternative food in the absence of prey. In addition, the prey population can attain their carrying capacity level when predators go to extinction. We demonstrate that the solutions are non‐negatives and bounded (dissipativity and permanence of population in many other works). Furthermore, some simulations to reinforce our mathematical results are shown, and we further discuss their ecological meanings. Copyright © 2017 John Wiley & Sons, Ltd.     

MULTISPECIES AND SINGLE‐SPECIES MODELS OF FISH POPULATION DYNAMICS: COMPARING PARAMETER ESTIMATES

Abstract Population features inferred from single‐species, age‐structured models are compared to those inferred from a multispecies, age‐structured model that includes predator‐prey interactions among three commercially harvested fish species—walleye pollock, Atka mackerel, and Pacific cod—on the Aleutian Shelf, Alaska. The multispecies framework treats the single‐species models and data as a special case of the multispecies model and data. The same data from fisheries and surveys are used to estimate model parameters for both single‐species and multispecies configurations of the model. Additionally, data from stomach samples and predator rations are used to estimate the parameters of the multispecies model. One form of the feeding functional response, predator pre‐emption, was selected using AIC from seven alternative models for how the predation rate changes with the densities of prey and possibly other predators. Differences in estimated population dynamics and productivity between the multispecies and single‐species models were observed. The multispecies model estimated lower mackerel population sizes from 1964–2003 than the single‐species model, while the spawning biomass of pollock was estimated to have declined more than three times faster since 1964 by the multispecies model. The variances around the estimates of spawning biomass were smaller for mackerel and larger for pollock in the multispecies model compared to the single‐species model.     

Periodic solutions in non autonomous predator prey system with delays

In this paper, we consider a biological model for two predators and one prey with periodic delays. By assuming that one predator consumes prey according to Holling II functional response while the other predator consumes prey according to the Beddington–DeAngelis functional response, based on the coincidence degree theory, the existence of positive periodic solutions for this model is obtained under suitable conditions.     

The impact of Allee effect on a predator–prey system with Holling type II functional response

In this paper, the Allee effect is incorporated into a predator–prey model with Holling type II functional response. Compared with the predator–prey model without Allee effect, we find that the Allee effect of prey species increases the extinction risk of both predators and prey. When the handling time of predators is relatively short and the Allee effect of prey species becomes strong, both predators and prey may become extinct. Moreover, it is shown that the model with Allee effect undergoes the Hopf bifurcation and heteroclinic bifurcation. The Allee effect of prey species can lead to unstable periodical oscillation. It is also found that the positive equilibrium of the model could change from stable to unstable, and then to stable when the strength of Allee effect or the handling time of predators increases continuously from zero, that is, the model admits stability switches as a parameter changes. When the Allee effect of prey species becomes strong, longer handling time of predators may stabilize the coexistent steady state.     

Positive periodic solutions for Lotka–Volterra systems with a general attack rate

The paper deals with a non-autonomous Lotka–Volterra type system, which in particular may include logistic growth of the prey population and hunting cooperation between predators. We focus on the existence of positive periodic solutions by using an operator approach based on the Krasnosel’skii homotopy expansion theorem. We give sufficient conditions in order that the localized periodic solution does not reduce to a steady state. Particularly, two typical expressions for the functional response of predators are discussed.     

On non-selective harvesting of a multispecies fishery

The present paper deals with the problem of non-selective harvesting of a prey-predator system in which both the prey and the predator species obey the law of logistic growth and each predators functional response to the prey approaches a constant as the prey population increases. Boundedness of the exploited system is examined. The existence of its steady states and their stability are studied using eigenvalue analysis. The existence of bionomic equilibria has been considered. The problem of determining the optimal harvest policy is then solved by using Pontryagin's maximal principle. Finally, some numerical examples are given to illustrate the results.     

Coexistence and stability of predator-prey model with Beddington-DeAngelis functional response and stage structure

We present a predator-prey model of Beddington-DeAngelis type functional response with stage structure on prey. The constant time delay is the time taken from birth to maturity about the prey. By the uniform persistence theories and monotone dynamic theories, sharp threshold conditions which are both necessary and sufficient for the permanence and extinction of the model as well as the sufficient conditions for the global stability of the coexistence equilibria are obtained. Biologically, it is proved that the variation of prey stage structure can affect the permanence of the system and drive the predator into extinction by changing the prey carrying capacity: Our results suggest that the predator coexists with prey permanently if and only if predator's recruitment rate at the peak of prey abundance is larger than its death rate; and that the predator goes extinct if and only if predator's possible highest recruitment rate is less than or equal to its death rate; furthermore, our results also show that a sufficiently large mutual interference by predators can stabilize the system.     

Existence of global solutions for a three‐species predator‐prey model with cross‐diffusion

In this short note, we study a strongly coupled system of partial differential equations which models the dynamics of a two‐predator‐one‐prey ecosystem in which the prey exercises defense switching and the predators collaboratively take advantage of the prey's strategy. We prove the existence of global strong solutions. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)