Maximum sustainable yield and species extinction in a prey–predator system: some new results

Though the maximum sustainable yield (MSY) approach has been legally adopted for the management of world fisheries, it does not provide any guarantee against from species extinction in multispecies communities. In the present article, we describe the appropriateness of the MSY policy in a Holling–Tanner prey–predator system with different types of functional responses. It is observed that for both type I and type II functional responses, harvesting of either prey or predator species at the MSY level is a sustainable fishing policy. In the case of combined harvesting, both the species coexist at the maximum sustainable total yield (MSTY) level if the biotic potential of the prey species is greater than a threshold value. Further, increase of the biotic potential beyond the threshold value affects the persistence of the system.     

Analysis of optimal harvesting of a predator-prey model with Holling type IV functional response

This paper investigates a series of harvesting problems of a harvested predator-prey system with Holling type IV functional response. The bionomic equilibrium, the maximum sustainable total yield (MSTY) and the optimal economic profit of the proposed system are studied. It is proved that the MSTY does not exist under the independent harvesting mode, while it may be found with the same predator and prey harvesting efforts mode. By applying the Bang-Bang control and the singular control to the harvesting strategy, the optimal equilibrium state of the discussed system converges faster than that of the system utilizing a single harvesting strategy with the fixed harvesting effort. The MISER 3 software package is adopted to obtain the optimization schemes of two control problems by using the control parameterization method. The findings of our study provide a theoretical basis for biological resource management.     

Context-dependent effects of predator removal from experimental microcosm communities

The loss of a predator from an ecological community can cause large changes in community structure and ecosystem processes, or have very little consequence for the remaining species and ecosystem. Understanding when and why the loss of a predator causes large changes in community structure and ecosystem processes is critical for understanding the functional consequences of biodiversity loss. We used experimental microbial communities to investigate how the removal of a large generalist predator affected the extinction frequency, population abundance and total biomass of its prey. We removed this predator in the presence or absence of an alternative, more specialist, predator in order to determine whether the specialist predator affected the outcome of the initial species removal. Removal of the large generalist predator altered some species' populations but many were unaffected and no secondary extinctions were observed. The specialist predator, though rare, altered the response of the prey community to the removal of the large generalist predator. In the absence of the specialist predator, the effects of the removal were only measurable at the level of individual species. However, when the specialist predator was present, the removal of the large generalist predator affected the total biomass of prey species. The results demonstrate that the effect of species loss from high trophic levels may be very context-dependent, as rare species can have disproportionately large effects in food webs.     

Dynamics of a intraguild predation model with generalist or specialist predator

Intraguild predation (IGP) is a combination of competition and predation which is the most basic system in food webs that contains three species where two species that are involved in a predator/prey relationship are also competing for a shared resource or prey. We formulate two intraguild predation (IGP: resource, IG prey and IG predator) models: one has generalist predator while the other one has specialist predator. Both models have Holling-Type I functional response between resource-IG prey and resource-IG predator; Holling-Type III functional response between IG prey and IG predator. We provide sufficient conditions of the persistence and extinction of all possible scenarios for these two models, which give us a complete picture on their global dynamics. In addition, we show that both IGP models can have multiple interior equilibria under certain parameters range. These analytical results indicate that IGP model with generalist predator has “top down” regulation by comparing to IGP model with specialist predator. Our analysis and numerical simulations suggest that: (1) Both IGP models can have multiple attractors with complicated dynamical patterns; (2) Only IGP model with specialist predator can have both boundary attractor and interior attractor, i.e., whether the system has the extinction of one species or the coexistence of three species depending on initial conditions; (3) IGP model with generalist predator is prone to have coexistence of three species.     

Balancing yield with resilience and conservation objectives in harvested predator–prey communities

The global overexploitation of fish stocks is endangering many marine food webs. Scientists and managers now call for an ecosystem‐based fisheries management, able to take into account the complexity of marine ecosystems and the multiple ecosystem services they provide. By contrast, many fishery management plans only focus on maximizing the productivity of harvested stocks. Such practices are suggested to affect other ecosystem services, altering the integrity and resilience of natural communities. Here we show that while yield‐maximizing policies can allow for coexistence and resilience in predator–prey communities, they are not optimal in a multi‐objective context. We find that although total prey and predator maximum yields are higher with a prey‐oriented harvest, focusing on the predator improves species coexistence. Also, moderate harvesting of the predator can enhance resilience. Furthermore, increasing maximum yields by changing catchabilities improves resilience in predator‐oriented systems, but reduces it in prey‐oriented systems. In a multi‐objective context, optimal harvesting strategies involve a general tradeoff between yield and resilience. Resilience‐maximizing strategies are however compatible with quite high yields, and should often be favored. Our results further suggest that balancing harvest between trophic levels is often best at maintaining simultaneously species coexistence, resilience and yield.     

Bistability and limit cycles in generalist predator–prey dynamics

Since generalist predators feed on a variety of prey species they tend to persist in an ecosystem even if one particular prey species is absent. Predation by generalist predators is typically characterized by a sigmoidal functional response, so that predation pressure for a given prey species is small when the density of that prey is low. Many mathematical models have included a sigmoidal functional response into predator–prey equations and found the dynamics to be more stable than for a Holling type II functional response. However, almost none of these models considers alternative food sources for the generalist predator. In particular, in these models, the generalist predator goes extinct in the absence of the one focal prey. We model the dynamics of a generalist predator with a sigmoidal functional response on one dynamic prey and fixed alternative food source. We find that the system can exhibit up to six steady states, bistability, limit cycles and several global bifurcations.     

Numerical and dietary responses of a predator community in a temperate zone of Europe

The generalist predation hypothesis predicts that the functional responses of generalist predator species should be quicker than those of specialist predators and have a regulating effect on vole populations. New interpretations of their role in temperate ecosystems have, however, reactivated a debate suggesting generalist predators may have a destabilizing effect under certain conditions (e.g. landscape homogeneity, low prey diversity, temporary dominance of 1 prey species associated with a high degree of dietary specialization). We studied a rich predator community dominated by generalist carnivores ( Martes spp., Vulpes vulpes, Felis catus ) over a 6 yr period in farmland and woodland in France. The most frequent prey were small rodents (mostly Microtus arvalis , a grassland species, and Apodemus spp., a woodland species). Alternative prey were diverse and dominated by lagomorphs ( Oryctolagus cuniculus, Lepus europeus ). We detected a numerical response among specialist carnivores but not among generalist predators. The dietary responses of generalist predators were fairly complex and most often dependent on variation in density of at least 1 prey species. These results support the generalist predation hypothesis. We document a switch to alternative prey, an increase of diet diversity, and a decrease of diet overlap between small and medium-sized generalists during the low density phase of M. arvalis . In this ecosystem, the high density phases of small mammal species are synchronous and cause a temporary specializing of several generalist predator species. This rapid functional response may indicate the predominant role of generalists in low amplitude population cycles of voles observed in some temperate areas.     

Biological Conservation of a Prey–Predator System Incorporating Constant Prey Refuge Through Provision of Alternative Food to Predators: A Theoretical Study

We describe a prey–predator system incorporating constant prey refuge through provision of alternative food to predators. The proposed model deals with a problem of non-selective harvesting of a prey–predator system in which both the prey and the predator species obey logistic law of growth. The long-run sustainability of an exploited system is discussed through provision of alternative food to predators. We have analyzed the variability of the system in presence of constant prey refuge and examined the stabilizing effect on predator–prey system. The steady states of the system are derived and dynamical behavior of the system is extensively analyzed around steady states. The optimal harvesting policy is formulated and solved with the help of Pontryagin’s maximal principle. Our objective is to maximize the monetary social benefit through protecting the predator species from extinction, keeping the ecological balance. Results finally illustrated with the help of numerical examples.     

Bioeconomic harvesting of a prey–predator fishery

This paper deals with the problem of non-selective harvesting of a prey–predator system by using a reasonable catch-rate function instead of usual catch-per-unit-efforthypothesis. Here both the prey and the predator species obey the law of logistic growth. We have taken the predator functional response to prey density in such a form that each predator's functional response to the prey density approaches a constant as the prey population increases. Boundedness of the exploited system is examined. The existence of its steady states and their stability (local and global) are studied using Eigenvalue analysis. The existence of bionomic equilibria has been illustrated using a numerical example. The problem of determining the optimal harvesting policy is then solved by using Pontryagin's maximum principle.     

The influence of generalist predators in spatially extended predator–prey systems

The presence of generalist predators is known to have important ecological impacts in several fields. They have wide applicability in the field of biological control. However, their role in the spatial distribution of predator and prey populations is still not clear. In this paper, the spatial dynamics of a predator–prey system is investigated by considering two different types of generalist predators. In one case, it is considered that the predator population has an additional food source and can survive in the absence of the prey population. In the other case, the predator population is involved in intraguild predation, i.e., the source of the additional food of the predator coincides with the food source of the prey population and thus both prey and predator populations compete for the same resource. The conditions for linear stability and Turing instability are analyzed for both the cases. In the presence of generalist predators, the system shows different pattern formations and spatiotemporal chaos which has important implications for ecosystem functioning not only in terms of their predictability, but also in influencing species persistence and ecosystem stability in response to abrupt environmental changes. This study establishes the importance of the consideration of spatial dynamics while determining optimal strategies for biological control through generalist predators.     

Enrichment can damp population cycles: a balance of inflexible and flexible interactions

Destabilization of one predator–one prey systems with an increase in nutrient input has been viewed as a paradox. We report that enrichment can damp population cycles by a food‐web structure that balances inflexible and flexible interaction links (i.e. specialist and generalist predators). We modeled six predator–prey systems involving three or four species in which the predators practice optimal foraging based on prey profitability determined by handling time. In all models, the balance of interaction links simultaneously decreased the amplitude of population oscillations and increased the minimum density with increasing enrichment, leading to a potential theoretical resolution of the paradox of enrichment in non‐equilibrium dynamics. The stabilization mechanism was common to all of the models. Important previous studies on the stability of food webs have also demonstrated that a balance of interaction strengths stabilizes systems, suggesting a general rule of ecosystem stability.     

Bioeconomic harvesting of a prey-predator fishery

This paper deals with the problem of non-selective harvesting of a prey-predator system by using a reasonable catch-rate function instead of usual catch-per-unit-efforthypothesis. Here both the prey and the predator species obey the law of logistic growth. We have taken the predator functional response to prey density in such a form that each predator's functional response to the prey density approaches a constant as the prey population increases. Boundedness of the exploited system is examined. The existence of its steady states and their stability (local and global) are studied using Eigenvalue analysis. The existence of bionomic equilibria has been illustrated using a numerical example. The problem of determining the optimal harvesting policy is then solved by using Pontryagin's maximum principle.     

Antipredator responses in Tetranychus urticae differ with predator specialization

The behavioural response of Tetranychus urticae to chemical cues from specialist predatory mites, Phytoseiulus persimilis, or generalist predatory bugs, Orius majusculus, on either bean or strawberry was studied in experimental arenas. Predators were placed on the leaf disc for 24 h and removed before T. urticae females were introduced. After 24 h, prey fecundity (number of eggs laid) and dispersal (number of prey drowned in the water barrier) were assessed. Chemical cues from the specialist predator resulted in reduced prey fecundity, significantly different from the generalist predator and control treatments. No interaction effect was found between plant species and prey fecundity, while significantly more eggs were laid on bean than on strawberry. Predator cues irrespective of predator specialization resulted in more prey dispersal than in the control. Findings emphasize the importance of specialization in the predator species complex for the degree and type of antipredator responses and resulting biological control.     

Optimal harvesting of prey–predator system with interval biological parameters: A bioeconomic model

The paper presents the study of one prey one predator harvesting model with imprecise biological parameters. Due to the lack of precise numerical information of the biological parameters such as prey population growth rate, predator population decay rate and predation coefficients, we consider the model with imprecise data as form of an interval in nature. Many authors have studied prey–predator harvesting model in different form, here we consider a simple prey–predator model under impreciseness and introduce parametric functional form of an interval and then study the model. We identify the equilibrium points of the model and discuss their stabilities. The existence of bionomic equilibrium of the model is discussed. We study the optimal harvest policy and obtain the solution in the interior equilibrium using Pontryagin’s maximum principle. Numerical examples are presented to support the proposed model.     

Predators alter the scaling of diversity in prey metacommunities

Although predator effects on the number of locally coexisting species are well understood, there are few formal predictions of how these local predator effects influence patterns of prey diversity at larger spatial scales. Building on the theory of island biogeography, we develop a simple model that describes how predators can alter the scaling of diversity in prey metacommunities and compares the effects of generalist and specialist predators on regional prey diversity. Generalist predators, which consume prey randomly with respect to species identity, are predicted to reduce α‐diversity and increase β‐diversity thereby maintaining regional diversity (γ‐diversity). Alternatively, specialist predators, which filter out prey species intolerant of predators, are predicted to reduce bothα‐diversity andβ‐diversity by causing the same prey species to be extirpated in each locality, resulting in regional prey species extinctions and lower γ‐diversity. These distinct effects of generalist and specialist predators on prey diversity at different spatial scales are uniquely shaped by the extent of predation within those metacommunities. Overall, our model results make general predictions for how different types of predators can differentially affect prey diversity across spatial scales, allowing a more complete understanding of the possible implications of predator eradications or introductions for biodiversity.     

Paradox of marine protected areas: suppression of fishing may cause species loss

A number of fish and invertebrate stocks have been depleted by overexploitation in recent years. To address this, marine protected areas (MPAs) are often established to protect biodiversity and recover stocks. We analyzed the potential impact of establishing MPAs on marine ecosystems using mathematical models. We demonstrate that establishment of an MPA can sometimes result in a considerable decline, or even extinction, of a species. We focus on a prey–predator system in two patches, one exposed to fishing activity and the other protected (MPA). Our analyses reveal that the establishment of the MPA can cause a reduction in prey abundance, and even extinction of the prey. Such unintended consequences are more likely to occur if the predator species is a generalist and if the MPA is intended to protect only the predatory species. Further, a mobile predator that migrates adaptively rather than randomly is associated with a greater reduction in prey abundance.     

A resolution of the paradox of enrichment

Theoretical studies have shown a paradoxical destabilizing response of predator-prey ecosystems to enrichment, but there is the gap between the intuitive view of nature and this theoretical prediction. We studied a minimal predator-prey system (a two predator-two prey system) in which the paradox of enrichment pattern can vanish; the destabilization with enrichment is reversed, leading to stabilization (a decrease in the amplitude of oscillation of population densities). For resolution of the paradox, two conditions must be met: (1) the same prey species must be preferred as a dietary item by both predator species, creating the potential for high exploitative competition between the predator species, and (2), while both predators are assumed to select their diet in accordance with optimal diet utilization theory, one predator must be a specialist and the other a generalist. In this system, the presence of a less profitable prey species can cause the increase in population oscillation amplitudes associated with increasing enrichment to be suppressed via the optimal diet utilization of the generalist predator. The resulting stabilization is explained by the mitigating effect of the less profitable prey showing better population growth with increasing enrichment on the destabilization underlying the specialist predator and prey relation, thus resolving the paradox of enrichment.     

Generalist and specialist predators that mediate permanence in ecological communities

 General dynamic models of systems with two prey and one or two predators are considered. After rescaling the equations so that both prey have the same intrinsic rate of growth, it is shown that there exists a generalist predator that can mediate permanence if and only if there is a population density of a prey at which its per-capita growth rate is positive yet less than its competitor’s invasion rate. In particular, this result implies that if the outcome of competition between the prey is independent of initial conditions, then there exists a generalist predator that mediates permanence. On the other hand, if the outcome of competition is contingent upon initial conditions (i.e., the prey are bistable), then there may not exist a suitable generalist predator. For example, bistable prey modeled by the Ayala–Gilpin (θ-Logistic) equations can be stabilized if and only if θ<1 for one of the prey. It is also shown that two specialist predators always can mediate permanence between bistable prey by creating a repelling heteroclinic cycle consisting of fixed points and limit cycles. Received 10 August 1996; received in revised form 21 March 1997     

Asymmetric reproductive interference between specialist and generalist predatory ladybirds

1.?Closely related species often differ greatly in the quality and breadth of resources exploited, but the actual mechanisms causing these differences are poorly understood. Because in the laboratory specialized species often survive and perform as well or better on host species that are never utilized in nature, negative ecological interactions restricting host range must exist. Here, we focused on reproductive interference, which has been theoretically predicted to drive niche separation between closely related species with overlapping mating signals. 2.?We examined the interspecific sexual interactions in relation to ecological specialization and generalization in two sibling ladybird species, Harmonia yedoensis and Harmonia axyridis. Harmonia yedoensis is a specialist predator that preys only on pine aphids, which are highly elusive prey for ladybird hatchlings, whereas H.?axyridis is a generalist predator with a broad prey and habitat range. 3.?We experimentally showed that conspecific sperm fertilized the vast majority of eggs regardless of mating order (i.e. conspecific sperm precedence) when a female of H.?yedoensis or H.?axyridis mated with both a conspecific and a heterospecific male. Moreover, we demonstrated that mating opportunities of H.?yedoensis females strongly decreased as heterospecific density increased relative to conspecific density. In contrast, in H.?axyridis, female mating success was high regardless of conspecific or heterospecific density. 4.?Our results suggest that the generalist H.?axyridis should be dominant to the specialist H.?yedoensis in terms of reproductive interference. Our results support the hypothesis that asymmetric reproductive interference from the dominant species may force the non-dominant species to become a specialist predator that exclusively utilizes less preferred prey in nature.     

Stochastic resonance in climate reddening increases the risk of cyclic ecosystem extinction via phase-tipping

Human activity is leading to changes in the mean and variability of climatic parameters in most locations around the world. The changing mean has received considerable attention from scientists and climate policy makers. However, recent work indicates that the changing variability, that is, the amplitude and the temporal autocorrelation of deviations from the mean, may have greater and more imminent impact on ecosystems. In this paper, we demonstrate that changes in climate variability alone could drive cyclic predator–prey ecosystems to extinction via so-called phase-tipping (P-tipping), a new type of instability that occurs only from certain phases of the predator–prey cycle. We construct a mathematical model of a variable climate and couple it to two self-oscillating paradigmatic predator–prey models. Most importantly, we combine realistic parameter values for the Canada lynx and snowshoe hare with actual climate data from the boreal forest. In this way, we demonstrate that critically important species in the boreal forest have increased likelihood of P-tipping to extinction under predicted changes in climate variability, and are most vulnerable during stages of the cycle when the predator population is near its maximum. Furthermore, our analysis reveals that stochastic resonance is the underlying mechanism for the increased likelihood of P-tipping to extinction.