Impacts of Biotic Resource Enrichment on a Predator–Prey Population

The environmental carrying capacity is usually assumed to be fixed quantity in the classical predator–prey population growth models. However, this assumption is not realistic as the environment generally varies with time. In a bid for greater realism, functional forms of carrying capacities have been widely applied to describe varying environments. Modelling carrying capacity as a state variable serves as another approach to capture the dynamical behavior between population and its environment. The proposed modified predator–prey model is based on the ratio-dependent models that have been utilized in the study of food chains. Using a simple non-linear system, the proposed model can be linked to an intra-guild predation model in which predator and prey share the same resource. Distinct from other models, we formulate the carrying capacity proportional to a biotic resource and both predator and prey species can directly alter the amount of resource available by interacting with it. Bifurcation and numerical analyses are presented to illustrate the system’s dynamical behavior. Taking the enrichment parameter of the resource as the bifurcation parameter, a Hopf bifurcation is found for some parameter ranges, which generate solutions that posses limit cycle behavior.     

Multiparameter bifurcation diagrams in predator-prey models with time lag

A predator-prey model is considered in which prey is limited by the carrying capacity of the environment, and predator growth rate depends on past quantities of prey. Conditions for stability of an equilibrium, and its bifurcation are established taking into account all the parameters.     

The subcritical collapse of predator populations in discrete-time predator-prey models.

Many discrete-time predator-prey models possess three equilibria, corresponding to (1) extinction of both species, (2) extinction of the predator and survival of the prey at its carrying capacity, or (3) coexistence of both species. For a variety of such models, the equilibrium corresponding to coexistence may lose stability via a Hopf bifurcation, in which case trajectories approach an invariant circle. Alternatively, the equilibrium may undergo a subcritical flip bifurcation with a concomitant crash in the predator's population. We review a technique for distinguishing between subcritical and supercritical flip bifurcations and provide examples of predator-prey systems with a subcritical flip bifurcation.     

Effects of behavioral adaptation on a predator-prey model

The Volterra-Lotka predator-prey equations are modified so that the predator's ability to utilize the prey varies in proportion to the average number of encounters between the two species in the past. The behavior of this adaptive system is then described in terms of three parameters — the carrying capacity of the prey, the relative death rate of the predator, and the predator's memoryspan. The most stable situation is shown to occur when the carrying capacity of the prey is large, the predator's death rate is close to zero, and the predator is able to adapt quickly to changing levels of prey density.     

Modeling the Effect of Prey Refuge on a Ratio-Dependent Predator–Prey System with the Allee Effect

The extinction of species is a major threat to the biodiversity. The species exhibiting a strong Allee effect are vulnerable to extinction due to predation. The refuge used by species having a strong Allee effect may affect their predation and hence extinction risk. A mathematical study of such behavioral phenomenon may aid in management of many endangered species. However, a little attention has been paid in this direction. In this paper, we have studied the impact of a constant prey refuge on the dynamics of a ratio-dependent predator–prey system with strong Allee effect in prey growth. The stability analysis of the model has been carried out, and a comprehensive bifurcation analysis is presented. It is found that if prey refuge is less than the Allee threshold, the incorporation of prey refuge increases the threshold values of the predation rate and conversion efficiency at which unconditional extinction occurs. Moreover, if the prey refuge is greater than the Allee threshold, situation of unconditional extinction may not occur. It is found that at a critical value of prey refuge, which is greater than the Allee threshold but less than the carrying capacity of prey population, system undergoes cusp bifurcation and the rich spectrum of dynamics exhibited by the system disappears if the prey refuge is increased further.     

Uniform persistence for sigmoidal diet selection with keystone prey species

In this paper we discuss uniform persistence (UP) criteria of two prey- one predator systems, where we consider that the predator's diet selection is a sigmoidal function of the most profitable prey type in place of a step function of conventional diet choice theory. We also derive UP results of the system with direct interspecific competition between the prey. The role of the most profitable prey item as a keystone species, the magnitude of its carrying capacity, the ability to withstand predation of both prey species, and the ratios of their profitability values (to predators) are important to whether or not adaptive foraging may promote UP. In general, foraging decision rules play no role in UP if the alternative prey item is the keystone species. The result is also not affected by the effect of direct competitive coexistence or dominance relationship of the prey. In some cases, dominance of one of the prey species provides the most advantageous situation for ensuring UP. Received: 1 February 1999 / Revised version: 20 September 1999 / Published online: 4 July 2000     

Effect of resource subsidies on predator–prey population dynamics: a mathematical model

The influence of a resource subsidy on predator–prey interactions is examined using a mathematical model. The model arises from the study of a biological system involving arctic foxes (predator), lemmings (prey), and seal carcasses (subsidy). In one version of the model, the predator, prey and subsidy all occur in the same location; in a second version, the predator moves between two patches, one containing only the prey and the other containing only the subsidy. Criteria for feasibility and stability of the different equilibrium states are studied both analytically and numerically. At small subsidy input rates, there is a minimum prey carrying capacity needed to support both predator and prey. At intermediate subsidy input rates, the predator and prey can always coexist. At high subsidy input rates, the prey cannot persist even at high carrying capacities. As predator movement increases, the dynamic stability of the predator–prey-subsidy interactions also increases.     

PERSISTENCE OF A THREE INTERACTING PREY-PREDATOR MODEL WITH REFUGES

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Opportunistic feeding by lions: non-preferred prey comprise an important part of lion diets in a habitat where preferred prey are abundant

Optimal foraging theory predicts less diverse predator diets with a greater availability of preferred prey. This narrow diet niche should then be dominated by preferred prey, with implications for predator–prey dynamics and prey population ecology. We investigated lion (Panthera leo) diets in Hluhluwe–iMfolozi Park (HiP), South Africa, to assess whether lions in a site with a high density of preferred prey (prey species weighing 92–632 kg as estimated from a published meta-analysis) have a narrow diet, consisting primarily of preferred prey. HiP is a useful study site to investigate this prediction because it is a productive landscape (with a high density of prey) where lion-preferred prey constitutes up to 33% of the prey available to lions. Furthermore, to investigate whether lions in HiP exhibit sex-specific diets as documented in other southern African populations, we estimated male and female lion diets separately. We were specifically interested in testing whether traditional approaches of estimating lion diets at the population level mask sex-specific predation patterns, with possible implications for management of lions in small to medium-sized fenced reserves. Lions in HiP preferred larger prey species (63–684 kg) and had diets with a larger proportion of preferred prey than reported in an African-wide meta-analysis. However, despite the high density of preferred prey species, 36% of lion diets still consisted of typically non-preferred species such as nyala (Tragelaphus angasii). This finding suggests that lions in HiP maintain a degree of opportunism even when preferred prey are abundant. Therefore, abundant, non-preferred prey are likely to be an important resource for lion populations. Sex-specific differences in lion diets were evident in HiP, suggesting that estimation of lion resource use and carrying capacity should consider opportunistic hunting and sex-specific differences in lion diets.     

Consequences of symbiosis for food web dynamics

Basic Lotka-Volterra type models in which mutualism (a type of symbiosis where the two populations benefit both) is taken into account, may give unbounded solutions. We exclude such behaviour using explicit mass balances and study the consequences of symbiosis for the long-term dynamic behaviour of a three species system, two prey and one predator species in the chemostat. We compose a theoretical food web where a predator feeds on two prey species that have a symbiotic relationships. In addition to a species-specific resource, the two prey populations consume the products of the partner population as well. In turn, a common predator forages on these prey populations. The temporal change in the biomass and the nutrient densities in the reactor is described by ordinary differential equations (ODE). Since products are recycled, the dynamics of these abiotic materials must be taken into account as well, and they are described by odes in a similar way as the abiotic nutrients. We use numerical bifurcation analysis to assess the long-term dynamic behaviour for varying degrees of symbiosis. Attractors can be equilibria, limit cycles and chaotic attractors depending on the control parameters of the chemostat reactor. These control parameters that can be experimentally manipulated are the nutrient density of the inflow medium and the dilution rate. Bifurcation diagrams for the three species web with a facultative symbiotic association between the two prey populations, are similar to that of a bi-trophic food chain; nutrient enrichment leads to oscillatory behaviour. Predation combined with obligatory symbiotic prey-interactions has a stabilizing effect, that is, there is stable coexistence in a larger part of the parameter space than for a bi-trophic food chain. However, combined with a large growth rate of the predator, the food web can persist only in a relatively small region of the parameter space. Then, two zero-pair bifurcation points are the organizing centers. In each of these points, in addition to a tangent, transcritical and Hopf bifurcation a global heteroclinic bifurcation is emanating. This heteroclinic cycle connects two saddle equilibria where the predator is absent. Under parameter variation the period of the stable limit cycle goes to infinity and the cycle tends to the heteroclinic cycle. At this global bifurcation point this cycle breaks and the boundary of the basin of attraction disappears abruptly because the separatrix disappears together with the cycle. As a result, it becomes possible that a stable two-nutrient–two-prey population system becomes unstable by invasion of a predator and eventually the predator goes extinct together with the two prey populations, that is, the complete food web is destroyed. This is a form of over-exploitation by the predator population of the two symbiotic prey populations. When obligatory symbiotic prey-interactions are modelled with Liebigs minimum law, where growth is limited by the most limiting resource, more complicated types of bifurcations are found. This results from the fact that the Jacobian matrix changes discontinuously with respect to a varying parameter when another resource becomes most limiting.Revised version: 21 July 2003     

Multiple predator effects result in risk reduction for prey across multiple prey densities

Investigating how prey density influences a prey’s combined predation risk from multiple predator species is critical for understanding the widespread importance of multiple predator effects. We conducted experiments that crossed six treatments consisting of zero, one, or two predator species (hellgrammites, greenside darters, and creek chubs) with three treatments in which we varied the density of mayfly prey. None of the multiple predator effects in our system were independent, and instead, the presence of multiple predator species resulted in risk reduction for the prey across both multiple predator combinations and all three levels of prey density. Risk reduction is likely to have population-level consequences for the prey, resulting in larger prey populations than would be predicted if the effects of multiple predator species were independent. For one of the two multiple predator combinations, the magnitude of risk reduction marginally increased with prey density. As a result, models predicting the combined risk from multiple predator species in this system will sometimes need to account for prey density as a factor influencing per-capita prey death rates.     

Effect of resource subsidies on predator-prey population dynamics: a mathematical model

The influence of a resource subsidy on predator-prey interactions is examined using a mathematical model. The model arises from the study of a biological system involving arctic foxes (predator), lemmings (prey), and seal carcasses (subsidy). In one version of the model, the predator, prey and subsidy all occur in the same location; in a second version, the predator moves between two patches, one containing only the prey and the other containing only the subsidy. Criteria for feasibility and stability of the different equilibrium states are studied both analytically and numerically. At small subsidy input rates, there is a minimum prey carrying capacity needed to support both predator and prey. At intermediate subsidy input rates, the predator and prey can always coexist. At high subsidy input rates, the prey cannot persist even at high carrying capacities. As predator movement increases, the dynamic stability of the predator-prey-subsidy interactions also increases.     

How to model the local interaction in the predator–prey system at slow diffusion in a heterogeneous environment?

We examine the nonlinear reaction–diffusion–advection equations to modeling of the predator–prey system under heterogeneous carrying capacity of the prey, and Holling type II functional response. When advection and diffusion fluxes are absent or small, we detect the discrepancy between the resource (carrying capacity) and species distributions. The large diffusion eliminates this effect. We propose a modification of the functional response coefficients to provide the correlation between species distribution and resource in both cases. The numerical simulation of several models both under small and moderate advection–diffusion fluxes is carried out.     

Impact of fear effect on the growth of prey in a predator-prey interaction model

Several field data and experiments on a terrestrial vertebrates exhibited that the fear of predators would cause a substantial variability of prey demography. Fear for predator population enhances the survival probability of prey population, and it can greatly reduce the reproduction of prey population. Based on the experimental evidence, we proposed and analyzed a prey-predator system introducing the cost of fear into prey reproduction with Holling type-II functional response. We investigate all the biologically feasible equilibrium points, and their stability is analyzed in terms of the model parameters. Our mathematical analysis exhibits that for strong anti-predator responses can stabilize the prey-predator interactions by ignoring the existence of periodic behaviors. Our model system undergoes Hopf bifurcation by considering the birth rate r₀ as a bifurcation parameter. For larger prey birth rate, we investigate the transition to a stable coexisting equilibrium state, with oscillatory approach to this equilibrium state, indicating that the greatest characteristic eigenvalues are actually a pair of imaginary eigenvalues with real part negative, which is increasing for r₀. We obtained the conditions for the occurrence of Hopf bifurcation and conditions governing the direction of Hopf bifurcation, which imply that the prey birth rate will not only influence the occurrence of Hopf bifurcation but also alter the direction of Hopf bifurcation. We identify the parameter regions associated with the extinct equilibria, predator-free equilibria and coexisting equilibria with respect to prey birth rate, predator mortality rates. Fear can stabilize the predator-prey system at an interior steady state, where all the species can exists together, or it can create the oscillatory coexistence of all the populations. We performed some numerical simulations to investigate the relationship between the effects of fear and other biologically related parameters (including growth/decay rate of prey/predator), which exhibit the impact that fear can have in prey-predator system. Our numerical illustrations also demonstrate that the prey become less sensitive to perceive the risk of predation with increasing prey growth rate or increasing predators decay rate.     

Amphipod (Gammarus minus) responses to predators and predator impact on amphipod density

Recent theoretical work suggests that predator impact on local prey density will be the result of interactions between prey emigration responses to predators and predator consumption of prey. Whether prey increase or decrease their movement rates in response to predators will greatly influence the impact that predators have on prey density. In stream systems the type of predator, benthic versus water-column, is expected to influence whether prey increase or decrease their movement rates. Experiments were conducted to examine the response of amphipods (Gammarus minus) to benthic and water-column predators and to examine the interplay between amphipod response to predators and predator consumption of prey in determining prey density. Amphipods did not respond to nor were they consumed by the benthic predator. Thus, this predator had no impact on amphipod density. In contrast, amphipods did respond to two species of water-column predators (the predatory fish bluegills, Lepomis macrochirus, and striped shiners, Luxilus chrysocephalus) by decreasing their activity rates. This response led to similar positive effects on amphipod density at night by both species of predatory fish. However, striped shiners did not consume many amphipods, suggesting their impact on the whole amphipod “population” was zero. In contrast, bluegills consumed a significant number of amphipods, and thus had a negative impact on the amphipod “population”. These results lend support to theoretical work which suggests that prey behavioral responses to predators can mask the true impact that predators have on prey populations when experiments are conducted at small scales. Received: 21 March 1997 / Accepted: 15 December 1997     

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.     

Diet composition in San Francisco Estuary striped bass: does trophic adaptability have its limits?

Trophic adaptability is a term used to describe feeding flexibility in fishes. Though a useful conceptual starting point, fishes often face constraints on their ability to switch prey that could limit feeding success even when prey switching is observed. We compared striped bass diet compositions summarized from previously published studies in California’s Sacramento-San Joaquin Delta during two time periods (1963–1964 and 2001–2003), which allowed us to evaluate trophic adaptability in San Francisco Estuary striped bass at multiple time scales, ranging from intra-annual to multidecadal. The Delta is the landward region of the San Francisco Estuary; over time between the study periods, the Delta underwent substantial changes in potential prey availability for striped bass. We found evidence for trophic adaptability in San Francisco Estuary (SFE) striped bass at all temporal scales examined. Despite this ability to adapt to changes in prey availability, the relative abundance and carrying capacity of young striped bass have declined. This decline has previously been associated with substantial declines in their dominant historical prey—mysid shrimp. Our results, coupled with these previous findings, indicate that trophic adaptability may have limited usefulness as a conceptual model to predict foraging success when other food web constraints are not considered. We speculate that this is particularly true in highly invaded ecosystems like the San Francisco Estuary because invading species often introduce substantial and permanent changes into food webs, decreasing the likelihood that a predator will find prey assemblages that fully replace historical prey assemblages.     

Comparing functional responses in predator-infected eco-epidemics models

The current paper deals with the mathematical models of predator–prey system where a transmissible disease spreads among the predator species only. Four mathematical models are proposed and analysed with several popular predator functional responses in order to show the influence of functional response on eco-epidemic models. The existence, boundedness, uniqueness of solutions of all the models are established. Mathematical analysis including stability and bifurcation are observed. Comparison among the results of these models allows the general conclusion that relevant behaviour of the eco-epidemic predator–prey system, including switching of stability, extinction, persistence and oscillations for any species depends on four important parameters viz. the rate of infection, predator interspecies competition and the attack rate on susceptible predator. The paper ends with a discussion of the biological implications of the analytical and numerical results.     

Predator–prey system with strong Allee effect in prey

Global bifurcation analysis of a class of general predator–prey models with a strong Allee effect in prey population is given in details. We show the existence of a point-to-point heteroclinic orbit loop, consider the Hopf bifurcation, and prove the existence/uniqueness and the nonexistence of limit cycle for appropriate range of parameters. For a unique parameter value, a threshold curve separates the overexploitation and coexistence (successful invasion of predator) regions of initial conditions. Our rigorous results justify some recent ecological observations, and practical ecological examples are used to demonstrate our theoretical work.