The stage-structured predator-prey model and optimal harvesting policy

In this paper, we establish a mathematical model of two species with stage structure and the relation of predator-prey, to obtain the necessary and sufficient condition for the permanence of two species and the extinction of one species or two species. We also obtain the optimal harvesting policy and the threshold of the harvesting for sustainable development.     

A comparison of two predator-prey models with Holling's type I functional response

In this paper, we analyze a laissez-faire predator-prey model and a Leslie-type predator-prey model with type I functional responses. We study the stability of the equilibrium where the predator and prey coexist by both performing a linearized stability analysis and by constructing a Lyapunov function. For the Leslie-type model, we use a generalized Jacobian to determine how eigenvalues jump at the corner of the functional response. We show, numerically, that our two models can both possess two limit cycles that surround a stable equilibrium and that these cycles arise through global cyclic-fold bifurcations. The Leslie-type model may also exhibit super-critical and discontinuous Hopf bifurcations. We then present and analyze a new functional response, built around the arctangent, that smoothes the sharp corner in a type I functional response. For this new functional response, both models undergo Hopf, cyclic-fold, and Bautin bifurcations. We use our analyses to characterize predator-prey systems that may exhibit bistability.     

The effect of the Holling type II functional response on apparent competition

This article analyzes the classical 2-resource-1-consumer apparent competition community module with the Holling type II functional response. Two types of resource regulation (top-down vs. combined top-down and bottom-up) and two types of consumer behaviors (inflexible consumers with fixed preferences for resources vs. adaptive consumers) are considered. When resources grow exponentially and consumers are inflexible foragers, one resource is always outcompeted due to strong apparent competition. Density dependent resource growth relaxes apparent competition so that resources can coexist. As multiple attractors (either equilibria or limit cycles) coexist, population dynamics and community composition depend on initial population densities. Population dynamics change dramatically when consumers forage adaptively. In this case, the results both for top-down, and combined top-down and bottom-up regulation are similar and they show that species persistence occurs for a much larger set of parameter values when compared with inflexible consumers. Moreover, population dynamics will be chaotic when resource carrying capacities are high enough. This shows that adaptive consumer switching can destabilize population dynamics.     

Optimal harvesting for a predator-prey metapopulation

In this paper we present a deterministic, discrete-time model for a two-patch predator-prey metapopulation. We study optimal harvesting for the metapopulation using dynamic programming. Some rules are established as generalizations of rules for a single-species metapopulation harvesting theory. We also establish rules to harvest relatively more (or less) vulnerable prey subpopulations and more (or less) efficient predator subpopulations.     

Modelling Holling type II functional response in deterministic and stochastic food chain models with mass conservation

The Rosenzweig-MacArthur predator-prey model is the building block in modeling food chain, food webs and ecosystems. There are a number of hidden assumptions involved in the derivation. For instance the prey population growth is logistic without predation but also with predation. In order to reveal these we will start with modelling a resource-predator-prey system in a closed spatially homogeneous environment. This allows us to keep track of the nutrient flow. With an instantaneous remineralisation of the products excreted in the environment by the populations and dead body mass there is conservation of mass. This allows for a model dimension reduction and yields the mass balance predator-prey model. When furthermore the searching and handling processes are much faster that the population changing rates, the trophic interaction is described by a Holling type II functional response, also assumed in the Rosenzweig-MacArthur model. The derivation uses an extended deterministic model with number of searching and handling predators as model variables where the ratio of the predator/prey body masses is used as a mechanistic time-scale parameter. This extended model is also used as a starting point for the derivation of a stochastic model. We will investigate the stochastic effects of random switching between searching and handling of the predators and predator dying. Prey growth by consumption of ambient resources is still deterministic and therefore the stochastic model is hybrid. The transient dynamics is studied by numerical Monte Carlo simulations and also the quasi-equilibrium distribution for the population quantities is calculated. The body mass of the prey individual is the scaling parameter in the stochastic model formulation. This allows for a quantification of the mean-field approximation criterion for the justification of replacement of the stochastic by a deterministic model.     

Analysis of a delayed diffusive model with Beddington-DeAngelis functional response

In this paper, a delayed diffusive phytoplankton-zooplankton model with Beddington DeAngelis functional response and toxins is investigated. Existence of equilibria of the system are solved. The global asymptotic stability of the zooplankton-free equilibrium is obtained. The local stability of the coexistent equilibrium and existence of Hopf bifurcation are discussed. In addition, the properties of the Hopf bifurcation are studied based on the center manifold and normal form theory for partial differential equations. Finally, some numerical simulations are also carried out to confirm our theoretical analysis.     

Emergence of Holling type III zooplankton functional response: Bringing together field evidence and mathematical modelling

Food-web population models are rather sensitive to parameterization of functional response in predation terms. Theoretical studies predict enhancing of ecosystems’ stability for a functional response of sigmoid type (Holling type III). The choice of a correct type of response is especially important for modelling outcome of grazing control of algal blooms by zooplankton in nutrient-rich ecosystems. Extensive experiments on zooplankton feeding in laboratories show non-sigmoid nature of response for most herbivorous zooplankton species. As a consequence, there is a strong opinion in literature that the implementation of Holling III type grazing in plankton models is biologically meaningless. I argue, however, that such an ‘evident’ claim might be wrong and sigmoid functional responses in real plankton communities would emerge more often than was suggested earlier. Especially, this concerns plankton models without vertical resolution, which ignore heterogeneity in vertical distribution of species. Having conducted extensive literature search of data on zooplankton feeding in situ, I show that vertical heterogeneity in food distribution as well as active food searching behaviour of zooplankton can modify the type of functional response. In particular, the rate of food intake by the whole zooplankton population in the column, as a function of total amount of food, often exhibits a sigmoid behaviour, instead of a non-sigmoid one postulated previously based on laboratory experiments. This conceptual discrepancy is due to the ability of zooplankton to feed mostly in layers with high algal density. I propose a generic model explaining the observed alteration of type between overall and local functional responses. I show that emergence of Holling type III in plankton systems is due to mechanisms different from those well known in the ecological literature (e.g. food search learning, existence of alternative food, refuge for prey).     

An analysis of an ordinary differential equation model for a two-species predator-prey system with harvesting and stocking

An analysis is presented for a model of a two-species predator-prey system where each species can be harvested or stocked. Using methods from bifurcation theory the qualitative nature of the steady-state solutions is examined. The effect of harvesting and stocking rates and the prey carrying capacity is examined in detail.     

Stability regions in predator-prey systems with constant-rate prey harvesting

Summary We analyze the global behavior of a predator-prey system, modelled by a pair of non-linear ordinary differential equations, under constant-rate prey harvesting. By methods analogous to those used to study predator harvesting, we characterize the theoretically possible structures and transitions. With the aid of a computer simulation we construct examples to show which of these transitions can be realized in a biologically plausible model.Sponsored by the United States Army under Contract No. DAAG29-75-C-0024 and the National Research Council of Canada, Grant No. 67-3138.     

Simple testing procedures for the Holling type II model

In this paper, we consider predator–prey data that can be viewed as solutions to a planar system of ordinary differential equations (ODE) observed with random error. The ODE system admits a limit cycle, while the random error is supposed to act additively in the log-scale. One of the oldest such systems is Holling’s type II model. In spite of its simplicity, it is still very popular in data analyses, although more sophisticated models have been introduced in the literature. We propose a simple way of deciding whether a set of predator–prey pairs is indicative or not of a departure from this basic model by exploiting the geometric properties of the solution in the phase plane. To illustrate our method, we use simulated and real data.     

The influence of the type 3 (sigmoid) functional response upon the stability of predator-prey difference models

Sigmoid functional responses are found to exert a stabilizing influence upon a discrete-generation predator-prey model in a way analogous to that found in continuous predator-prey models. The precise effect depends upon the degree to which a predator's feeding history influences its reproductive success. The time delay intrinsic in difference equation models imposes constraints not found in differential models, however, it is shown that in an otherwise unstable model the inclusion of a sigmoid functional response can result in local stability. With the addition of prey self-regulation the stabilizing influence of the functional response acts in concert with self-regulation, as it does in continuous models. These results show that the effect of the sigmoid response upon stability is not dependent upon the assumption of continuity, and reinforces the view that sigmoid responses could be an important factor stabilizing natural communities.     

Effect of delay in a Lotka-Volterra type predator-prey model with a transmissible disease in the predator species

We consider a system of delay differential equations modeling the predator-prey ecoepidemic dynamics with a transmissible disease in the predator population. The time lag in the delay terms represents the predator gestation period. We analyze essential mathematical features of the proposed model such as local and global stability and in addition study the bifurcations arising in some selected situations. Threshold values for a few parameters determining the feasibility and stability conditions of some equilibria are discovered and similarly a threshold is identified for the disease to die out. The parameter thresholds under which the system admits a Hopf bifurcation are investigated both in the presence of zero and non-zero time lag. Numerical simulations support our theoretical analysis.     

Permanence of a stage-structured predator-prey system with a class of functional responses

A stage-structured predator-prey system incorporating a class of functional responses is presented in this article. By analyzing the system and using the standard comparison theorem, the sufficient conditions are derived for permanence of the system and non-permanence of predators.     

On constant effort harvesting and stocking in a class of predator-prey systems

We analyze the global behaviour of predator-prey systems under constant effort harvesting and stocking of either or both species. Mathematically this is equivalent to the behaviour of an unharvested system with different parameters; in particular every solution tends either to an equilibrium or to a limit cycle, and there is no possibility of finite-time extinction of either species if both populations are positive initially. The dependence of the behaviour on the harvesting efforts is discussed.     

Effects of prey refuges on a predator-prey model with a class of functional responses: The role of refuges

In this paper, the effects of refuges used by prey on a predator-prey interaction with a class of functional responses are studied by using the analytical approach. The refuges are considered as two types: a constant proportion of prey and a fixed number of prey using refuges. We will evaluate the effects with regard to the local stability of the interior equilibrium point, the values of the equilibrium density and the long-term dynamics of the interacting populations. The results show that the effects of refuges used by prey increase the equilibrium density of prey population while decrease that of predators. It is also proved that the effects of refuges can stabilize the interior equilibrium point of the considered model, and destabilize it under a very restricted set of conditions which is disagreement with previous results in this field.     

Extinction and permanence of one-prey multi-predators of Holling type II function response system with impulsive biological control

In this paper, one investigates the dynamic behaviors of one-prey multi-predator model with Holling type II functional response by introducing impulsive biological control strategy (periodic releasing natural enemies at different fixed time). By using Floquet theorem and small amplitude perturbation method, it is proved that there exists an asymptotically stable pest-eradication periodic solution when the impulsive period is less than some critical value and permanence condition is established via the method of comparison involving multiple Liapunov functions. It is shown that multi-predator impulsive control strategy is more effective than the classical and single one.     

Analysis of a predator-prey system with predator switching

In this paper, we consider an interaction of prey and predator species where prey species have the ability of group defence. Thresholds, equilibria and stabilities are determined for the system of ordinary differential equations. Taking carrying capacity as a bifurcation parameter, it is shown that a Hopf bifurcation can occur implying that if the carrying capacity is made sufficiently large by enrichment of the environment, the model predicts the eventual extinction of the predator providing strong support for the so-called ‘paradox of enrichment’.     

A predator-prey model with infected prey

A predator-prey model with logistic growth in the prey is modified to include an SIS parasitic infection in the prey with infected prey being more vulnerable to predation. Thresholds are identified which determine when the predator population survives and when the disease remains endemic. For some parameter values the greater vulnerability of the infected prey allows the predator population to persist, when it would otherwise become extinct. Also the predation on the more vulnerable prey can cause the disease to die out, when it would remain endemic without the predators.     

Analysis of a predator-prey model with herd behavior and disease in prey incorporating prey refuge

In this work,we have introduced an eco-epidemiological model of an infected predator prey system.Incorporation of prey refuge gives that a fraction of the infected prey is available to the predator for consumption.Moreover,to make the model more realistic to the environment,we have introduced strong Allee effect in the susceptible population.Boundedness and positivity of the solution have been established.Local stability conditions of the equilibrium points have been found with the help of Routh-Hurwitz criterion and it has been observed that if a prey population is infected with a lethal disease,then both the prey(susceptible and infected)and predator cannot survive simultaneously in the system for any parametric values.The disease transmission rate and the attack rate on the susceptible have an important role to control the system dynamics.For different values of these two key parameters,we have got only healthy or disease-free or predation-free or a fluctuating disease-free or even a fluctuating predator-free system with some certain parametric conditions.