Carrying Capacity Of Prey Research Articles

DYNAMICS OF ANTI-PREDATOR BEHAVIOR AND EFFECT OF FEAR ON PREY–PREDATOR MODEL

Predator–prey interactions are the ubiquitous and natural phenomenon in an ecological system. Predators reduce the prey population’s density by direct killing, which is an essential part of any ecological system. Based on the experimental works, for overcoming predation pressure, prey uses a variety of mechanisms. With Holling type-II functional response, we examined a prey–predator system incorporating anti-predator behavior and the cost of fear into prey. Prey anti-predator activity is a counterattacking strategy in which adult prey targets adolescent predators in order to counteract the potential predation pressure. Fear of predation may disrupt the physiological state of prey species and lead to long loss of prey species. In this study, we investigated this aspect to use a dynamical modeling approach. This research finds a plethora of fascinating phenomena. The studied system exhibits a wide range of dynamics and bifurcations, including saddle-node, Hopf, homoclinic, and a Bogdanov–Takens bifurcation in co-dimension two are among the dynamics and bifurcations observed in the analyzed system. We performed some numerical simulations to investigate the effects of anti-predator behavior and fear on prey and found both affect the prey–predator dynamics significantly. Our numerical examples clearly show that as prey carrying capacity increases, so does the prey’s ability to perceive the risk of predation.

Conservation potentials and limitations of large carnivores in protected areas: A case study in Northeast China

AbstractProtected areas are considered the cornerstone of endangered wildlife conservation. However, quantified conservation potentials and limitations of large carnivores in protected areas are lacking. In the Northeast Tiger and Leopard National Park (NTLNP) in China, our camera trap survey in 2019 found 26–27 adult Amur tigers and 49–59 adult Amur leopards occurring in the park. Based on spatial area, current prey populations, and environmental carrying capacity of prey, we estimated the supportable number of tigers to be 55, 90, and 101 individuals, respectively. For leopard, these values were 95, 356, and 572, respectively. Further simulations indicated that human land use change scenarios did not contribute much to increasing the potential prey‐supportable populations of Amur tiger and leopard. Our results showed that the number of tigers and leopards in NTLNP is currently low and has a high recovery potential. However, even the highest supportable population is not enough to support the sustainable existence of an Amur tiger population. Therefore, we suggest that, in addition to further restoration and improvement of the prey population and habitat quality in NTLNP, managers should strengthen the connectivity between NTLNP and other habitat patches to form a well‐connected network of protected areas. Promoting the spread of tigers and leopards outwards from this source population in NTLNP through ecological corridor construction would enlarge the area of habitat and is a crucial measure for realizing the sustainable survival of an Amur tiger population in Northeast China.

Pick your trade-offs wisely: Predator-prey eco-evo dynamics are qualitatively different under different trade-offs

In recent decades, myriad studies have explored the population dynamics of coevolving populations of predator and prey. A variety of choices are made in these models: exponential or logistic prey growth in the absence of a predator, various forms of predator functional response, and uni- or bi-directional trait axes. In addition, some form of trade-offs are assumed in order to prevent run-away evolution of the prey and predator traits. While there is a considerable amount of theory regarding various forms of prey growth rates and predator functional responses, only a few studies have explored how different types of trade-offs affect predator-prey dynamics. Here, we compared two ditrophic coevolution models incorporating different trade-offs via dual effects of the prey trait on attack rate and either prey carrying capacity or intrinsic growth rate. We employed a standard dynamical systems approach to analyze the equilibrium conditions of each model and find conditions for non-equilibrium oscillatory coexistence. The exact effect of various parameters on the outcome of predator-prey interactions depends on whether the trade-offs affect the intrinsic growth rate or carrying capacity. In particular, coexistence is more likely when prey growth rate is affected by the evolving trait. In addition, in parameter regimes where cycles occur in both models, oscillations typically have larger periods and amplitudes when prey growth rate is affected by the evolving trait.

Noise-induced shifts in the population model with a weak Allee effect

We consider the Truscott–Brindley system of interacting phyto- and zooplankton populations with a weak Allee effect. We add a random noise to the parameter of the prey carrying capacity, and study how the noise affects the dynamic behavior of this nonlinear prey–predator model. Phenomena of the stochastic excitement and noise-induced shifts in zones of the Andronov–Hopf bifurcation and Canard explosion are analyzed on the base of the direct numerical simulation and stochastic sensitivity functions technique. A relationship of these phenomena with transitions between order and chaos is discussed.

Structural sensitivity and resilience in a predator–prey model with density-dependent mortality

Numerous formulations with the same mathematical properties can be relevant to model a biological process. Different formulations can predict different model dynamics like equilibrium vs. oscillations even if they are quantitatively close (structural sensitivity). The question we address in this paper is: does the choice of a formulation affect predictions on the number of stable states? We focus on a predator–prey model with predator competition that exhibits multiple stable states. A bifurcation analysis is realized with respect to prey carrying capacity and species body mass ratio within range of values found in food web models. Bifurcation diagrams built for two type-II functional responses are different in two ways. First, the kind of stable state (equilibrium vs. oscillations) is different for 26.0–49.4% of the parameter values, depending on the parameter space investigated. Using generalized modelling, we highlight the role of functional response slope in this difference. Secondly, the number of stable states is higher with Ivlev's functional response for 0.1–14.3% of the parameter values. These two changes interact to create different model predictions if a parameter value or a state variable is altered. In these two examples of disturbance, Holling's disc equation predicts a higher system resilience. Indeed, Ivlev's functional response predicts that disturbance may trap the system into an alternative stable state that can be escaped from only by a larger alteration (hysteresis phenomena). Two questions arise from this work: (i) how much complex ecological models can be affected by this sensitivity to model formulation? and (ii) how to deal with these uncertainties in model predictions?

Predator-prey dynamics with Allee effect in prey refuge

In this paper we establish a predator-prey model with a refuge and an open habitat for prey. The Allee effect in a prey refuge and the environment carrying capacity of prey are considered. According to biology of prey and predator, fast and slow time scales are considered in some parameters. Based on two different time scales, the system is divided into a fast system and a slow system. Applying the singular perturbation techniques, we analyze the dynamics on the slow system. The stability analyses are performed, and the Hopf bifurcation occurs when the environment carrying capacity of prey is greater than a critical value. This value is an increasing function of the Allee effect. By calculating the first Lyapunov coefficient, the stable periodical oscillation is shown. It is shown that the carrying capacity of prey and the Allee effect of prey in the refuge can influence biological environment.

Effects of harvesting and predator interference in a model of two-predators competing for a single prey

In the present paper we formulate a mathematical model of two predators living on a single biotic prey. The two predators interfere directly with each other. The predation function for the two predators are assumed to be different - one follows the mass action kinetics while the other obeys Holling type-II functional response. We also assume that one of the predators is economically viable and undergo harvesting at a rate proportional to its density. We perform local stability analysis about the boundary states where one of the predator species is absent. It is seen that, the existence and/or extinction criteria of each predator is controlled by the predation rates of one or both the predators. The interior equilibrium point exhibits unstable oscillatory behavior in the local analysis. We also perform a global analysis using Lyapunov function and LaSalle’s principle which shows global stability around this point for limited carrying capacity of prey. Finally we model the phenomenon of random harvesting by perturbing the harvest rate using Gaussian white noise. The resulting stochastic differential equation model is analyzed for exponential mean square stability which is shown to depend on the harvesting effort. Numerical simulation study of the model equations are also carried out.

Revisiting the Stability of Spatially Heterogeneous Predator-Prey Systems Under Eutrophication.

We employ partial integro-differential equations to model trophic interaction in a spatially extended heterogeneous environment. Compared to classical reaction-diffusion models, this framework allows us to more realistically describe the situation where movement of individuals occurs on a faster time scale than on the demographic (population) time scale, and we cannot determine population growth based on local density. However, most of the results reported so far for such systems have only been verified numerically and for a particular choice of model functions, which obviously casts doubts about these findings. In this paper, we analyse a class of integro-differential predator-prey models with a highly mobile predator in a heterogeneous environment, and we reveal the main factors stabilizing such systems. In particular, we explore an ecologically relevant case of interactions in a highly eutrophic environment, where the prey carrying capacity can be formally set to 'infinity'. We investigate two main scenarios: (1) the spatial gradient of the growth rate is due to abiotic factors only, and (2) the local growth rate depends on the global density distribution across the environment (e.g. due to non-local self-shading). For an arbitrary spatial gradient of the prey growth rate, we analytically investigate the possibility of the predator-prey equilibrium in such systems and we explore the conditions of stability of this equilibrium. In particular, we demonstrate that for a Holling type I (linear) functional response, the predator can stabilize the system at low prey density even for an 'unlimited' carrying capacity. We conclude that the interplay between spatial heterogeneity in the prey growth and fast displacement of the predator across the habitat works as an efficient stabilizing mechanism. These results highlight the generality of the stabilization mechanisms we find in spatially structured predator-prey ecological systems in a heterogeneous environment.

Modelling the dynamics of traits involved in fighting-predators-prey system.

We study the dynamics of a predator–prey system where predators fight for captured prey besides searching for and handling (and digestion) of the prey. Fighting for prey is modelled by a continuous time hawk–dove game dynamics where the gain depends on the amount of disputed prey while the costs for fighting is constant per fighting event. The strategy of the predator-population is quantified by a trait being the proportion of the number of predator-individuals playing hawk tactics. The dynamics of the trait is described by two models of adaptation: the replicator dynamics (RD) and the adaptive dynamics (AD). In the RD-approach a variant individual with an adapted trait value changes the population’s strategy, and consequently its trait value, only when its payoff is larger than the population average. In the AD-approach successful replacement of the resident population after invasion of a rare variant population with an adapted trait value is a step in a sequence changing the population’s strategy, and hence its trait value. The main aim is to compare the consequences of the two adaptation models. In an equilibrium predator–prey system this will lead to convergence to a neutral singular strategy, while in the oscillatory system to a continuous singular strategy where in this endpoint the resident population is not invasible by any variant population. In equilibrium (low prey carrying capacity) RD and AD-approach give the same results, however not always in a periodically oscillating system (high prey carrying-capacity) where the trait is density-dependent. For low costs the predator population is monomorphic (only hawks) while for high costs dimorphic (hawks and doves). These results illustrate that intra-specific trait dynamics matters in predator–prey dynamics.

Using Predator-Prey Theory to Predict Outcomes of Broadscale Experiments to Reduce Apparent Competition

Apparent competition is an important process influencing many ecological communities. We used predator-prey theory to predict outcomes of ecosystem experiments aimed at mitigating apparent competition by reducing primary prey. Simulations predicted declines in secondary prey following reductions in primary prey because predators consumed more secondary prey until predator numbers responded to reduced prey densities. Losses were exacerbated by a higher carrying capacity of primary prey and a longer lag time of the predator's numerical response, but a gradual reduction in primary prey was less detrimental to the secondary prey. We compared predictions against two field experiments where endangered woodland caribou (Rangifer tarandus caribou) were victims of apparent competition. First, when deer (Odocoileus sp.) declined suddenly following a severe winter, cougar (Puma concolor) declined with a 1-2-year lag, yet in the interim more caribou were killed by cougars, and caribou populations declined by 40%. Second, when moose (Alces alces) were gradually reduced using a management experiment, wolf (Canis lupus) populations declined but did not shift consumption to caribou, and the largest caribou subpopulation stabilized. The observed contrasting outcomes of sudden versus gradual declines in primary prey supported theoretical predictions. Combining theory with field studies clarified how to manage communities to mitigate endangerment caused by apparent competition that affects many taxa.

Predator interference and stability of predator-prey dynamics.

Predator interference, that is, a decline in the per predator consumption rate as predator density increases, is generally thought to promote predator-prey stability. Indeed, this has been demonstrated in many theoretical studies on predator-prey dynamics. In virtually all of these studies, the stabilization role is demonstrated as a weakening of the paradox of enrichment. With predator interference, stable limit cycles that appear as a result of environmental enrichment occur for higher values of the environmental carrying capacity of prey, and even a complete absence of the limit cycles can happen. Here we study predator-prey dynamics using the Rosenzweig-MacArthur-like model in which the Holling type II functional response has been replaced by a predator-dependent family which generalizes many of the commonly used descriptions of predator interference. By means of a bifurcation analysis we show that sufficiently strong predator interference may bring about another stabilizing mechanism. In particular, hysteresis combined with (dis)appearance of stable limit cycles imply abrupt increases in both the prey and predator densities and enhanced persistence and resilience of the predator-prey system. We encourage refitting the previously collected data on predator consumption rates as well as for conducting further predation experiments to see what functional response from the explored family is the most appropriate.

Modelling Effects of Rapid Evolution on Persistence and Stability in Structured Predator-Prey Systems

In this paper we explore the eco-evolutionary dynamics of a predator-prey model, where the prey population is structured according to a certain life history trait. The trait distribution within the prey population is the result of interplay between genetic inheritance and mutation, as well as selectivity in the consumption of prey by the predator. The evolutionary processes are considered to take place on the same time scale as ecological dynamics, i.e. we consider the evolution to be rapid. Previously published results show that population structuring and rapid evolution in such predator-prey system can stabilise an otherwise globally unstable dynamics even with an unlimited carrying capacity of prey. However, those findings were only based on direct numerical simulation of equations and obtained for particular parametrisations of model functions, which obviously calls into question the correctness and generality of the previous results. The main objective of the current study is to treat the model analytically and consider various parametrisations of predator selectivity and inheritance kernel. We investigate the existence of a coexistence stationary state in the model and carry out stability analysis of this state. We derive expressions for the Hopf bifurcation curve which can be used for constructing bifurcation diagrams in the parameter space without the need for a direct numerical simulation of the underlying integro-differential equations. We analytically show the possibility of stabilisation of a globally unstable predator-prey system with prey structuring. We prove that the coexistence stationary state is stable when the saturation in the predation term is low. Finally, for a class of kernels describing genetic inheritance and mutation we show that stability of the predator-prey interaction will require a selectivity of predation according to the life trait.

COEVOLUTIONARY DYNAMICS OF PREDATOR-PREY INTERACTIONS

In this paper, with the method of adaptive dynamics, we investigate the coevolution of phenotypic traits of predator and prey species. The evolutionary model is constructed from a deterministic approximation of the underlying stochastic ecological processes. Firstly, we investigate the ecological and evolutionary conditions that allow for continuously stable strategy and evolutionary branching. We find that evolutionary branching in the prey phenotype will occur when the frequency dependence in the prey carrying capacity is not strong. Furthermore, it is found that if the two prey branches move far away enough, the evolutionary branching in the prey phenotype will induce the secondary branching in the predator phenotype. The final evolutionary outcome contains two prey and two predator species. Secondly, we show that under symmetric interactions the evolutionary model admits a supercritical Hopf bifurcation if the frequency dependence in the prey carrying capacity is very weak. Evolutionary cycle is a likely outcome of the mutation-selection processes. Finally, we find that frequency-dependent selection can drive the predator population to extinction under asymmetric interactions.

EFEITOS DE PREFERÊNCIA ALIMENTAR E AUMENTO DE PRODUTIVIDADE NA DINÂMICA DE REDES TRÓFICAS

Diversity and species coexistence can emerge as a consequence of several factors such as, predator's prey preference, prey carrying capacity, predators' mutual interference. In fact, interactions among three or more species are strongly influenced by direct and indirect interactions. Moreover, food webs usually display an intricate tangle of interactions containing multiple prey, which in turn gives rise to several food items for predators besides allocthonous inputs. Within this context, predator's prey preference is commonly employed in mathematical models in order to investigate the interplay among food web structure, diversity, species coexistence and stability. A feature of these models is the categorization of the structure preference into two distinct forms: (1) fixed (“non-switching”); (2) variable (“switching”). The first one assumes that the predator has a fixed behavior with respect to the densities of its alternative prey, while the second is based on the relative densities of its alternative prey. This latter form is strictly related to the concept of predation risk -- prey are more susceptible to predation when their densities are more elevated than those of their counterparts. Given these frameworks, the effects of these two preference structures and the augmentation of primary productivity on the ultimate dynamics of some specific food webs will be reviewed in this work.

BIFURCATIONS, AND TEMPORAL AND SPATIAL PATTERNS OF A MODIFIED LOTKA–VOLTERRA MODEL

Bazykin proposed a Lotka–Volterra-type ecological model that accounts for simplified territoriality, which neither depends on territory size nor on food availability. In this study, we describe the global dynamics of the Bazykin model using analytical and numerical methods. We specifically focus on the effects of mutual predator interference and the prey carrying capacity since the variability of each could have especially dramatic ecological repercussions. The model displays a broad array of complex dynamics in space and time; for instance, we find the coexistence of a limit cycle and a steady state, and bistability of steady states. We also characterize super- and subcritical Poincaré–Andronov–Hopf bifurcations and a Bogdanov–Takens bifurcation. To illustrate the system's ability to naturally shift from stable to unstable dynamics, we construct bursting solutions, which depend on the slow dynamics of the carrying capacity. We also consider the stabilizing effect of the intraspecies interaction parameter, without which the system only shows either a stable steady state or oscillatory solutions with large amplitudes. We argue that this large amplitude behavior is the source of chaotic behavior reported in systems that use the MacArthur–Rosenzweig model to describe food-chain dynamics. Finally, we find the sufficient conditions in parameter space for Turing patterns and obtain the so-called "back-eye" pattern and localized structures.

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.

DO CHEAP TRIPS HELP BIRDS FEED THEIR CHICKS?

How costly is provisioning chicks and how fast should birds fly to do it?These are the questions Kyle Elliott and Anthony Gaston sought to answer by measuring the flight speeds of seabirds in the field. Over the past 30 or so years, numerous models have been developed to estimate the power requirements for flapping flight in birds. The models predict a U-shaped relationship between power requirements for flight and flight speed. From these models, the flight speeds requiring least power (Vmp) and giving maximum range (Vmr) can be calculated. From such models,R. Å. Norberg predicted in 1981 that birds feeding chicks should fly faster than Vmr to maximise energy delivery to nestlings. Elliott and Gaston sought to test this hypothesis by making field measurements of flight speeds of Brünnich's guillemot and northern fulmar before and after their chicks had been hatched.Guillemot chicks fledge before completing growth, which has been suggested to be due to the inability of parents to maintain the cost of provisioning. By contrast, it is thought that fulmars are under time, rather than energy,constraints. Elliott and Gaston hypothesised that both species should fly faster during chick rearing than during incubation to maximise chick growth rates. By using infra-red distance-sensing binoculars, they were able to time birds of both species over a distance of 850 m on a flyway into the breeding colony. The velocities measured during inbound and outbound flights were assumed to be representative of those during entire foraging trips.Guillemots were found not to differ in flight speed between incoming and outgoing flights and before and after their chicks hatched. By contrast,fulmars flew faster after their chicks hatched, and incoming flights were faster than outbound ones. guillemots, it seems, fly close to Vmr while fulmars fly closer to Vmp. Elliott and Gaston propose a number of explanations for possible causes of quantitative disagreement with predicted flight speeds. Fulmars have a`flatter' power curve than guillemots, so it is metabolically less costly for them to adjust flight speed. Furthermore, patterns of weight loss, and hence prey-carrying capacity, differ between the two species. The fulmar may lose 15% of its mass when brooding compared with during incubation, compared with 5% in the guillemot. This mass change might lead to a predicted decrease in flight speed close to that observed. As the authors point out, however, their estimates of power are dependent upon the assumption that birds make their entire outbound and inbound flights at speeds similar to those recorded in their measurement area close to the colony.This paper demonstrates that quantifiable differences in flight speeds are observed in some species of bird in response to different demands of chick provisioning and demonstrates that field measurements can go some way to validating the predictions made concerning the tradeoffs between power requirements, range and velocity in seabirds.

Predator-prey Interactions with Delays Due to Juvenile Maturation

This paper focuses on predator-prey models with juvenile/mature class structure for each of the predator and prey populations in turn, further classified by whether juvenile or mature individuals are active with respect to the predation process. These models include quite general prey recruitment at every stage of analysis, with mass action predation, linear predator mortality as well as delays in the dynamics due to maturation. As a base for comparison we briefly establish that the similar model without delays cannot support sustained oscillation, but it does have predator extinction or global approach to predator-prey coexistence depending on whether the ratio $\alpha $ of per predator predation at prey carrying capacity to the predator death rate is less than or greater than one. Our first model shows the effect of introducing an invulnerable juvenile prey class, appropriate, e.g., for some host-parasite interactions. In contrast our second model shows the effect of limiting predation to a prey juvenile class. Finally, in a third model we consider an inactive juvenile predator class, which would be appropriate for many traditional situations in which the generation time for the predator is significantly larger than that of the prey. In all cases the introduction of a juvenile class results in a system of three delay-differential equations from which the two equations for the mature class and the nonstructured class can be decoupled. We obtain some global stability results and identify a parameter $\alpha $, similar to the $\alpha $ of the unlagged model, which determines whether or not the predator is driven to extinction. With $\alpha >1$, and considering the maturation age of the juvenile class as a bifurcation parameter, we obtain Hopf bifurcations in our second and third models, while in the case of juvenile prey (in the first model) the unique coexistence equilibrium remains stable for all positive delays. Although the delay is "physically present" in all three models, we obtain scaled, nondimensional replacement models with that physical presence scaled out. After analyzing the scaled equations we show that all our results hold for the original models. We pursue the bifurcation in the inactive juvenile predator model with numerical simulations. Strikingly similar results over a variety of birth functions are observed. Increases of the maturation delay first produce Hopf bifurcation from steady state to periodic behavior. Even further increase in the delay produces instabilities of the bifurcating periodic solutions with corresponding interesting geometry in a two-dimensional plot of period vs. delay.

Seasonal subsidy stabilizes food web dynamics: Balance in a heterogeneous landscape

Resource subsidies from external habitats can substantially affect the food web dynamics of local habitats. In this paper, we explore a mathematical model that is tailored for a stream food web, studied by Nakano and colleagues, in which consumers, in situ prey and subsidies all show seasonal fluctuation. The model reveals that the food web dynamics are stabilized if subsidies increase in summer when in situ productivity is low. Consumer dynamics are stabilized because subsidies complement seasonal resource deficiency. In situ prey dynamics are stabilized because subsidies indirectly balance the predation pressure by consumers, with seasonal change in prey carrying capacity. In summer when prey carrying capacity is low, seasonally abundant subsidies indirectly decrease predation pressure, whereas in winter, with high prey carrying capacity, scarce subsidies increase the predation pressure. Our results suggest that temporal productivity differences between spatially linked habitats are important to promote the stability of food web dynamics in a landscape context.

A Dynamic Model of Cannibalism

A naive model of cannibalism assumes that it decreases intra-species competition, increasing per capita supply of resources. As a result, there is a cycle between prey population and carrying capacity of the environment, which oscillates around cannibals' population. The model is applied to examine the behaviour of firms in an oligopolistic industry. From this exercise emerges a general model of cannibalism which shows the economic rationale for the existence of cannibalistic strategies and has stronger properties than the naive model.