Predator-dependent Functional Response Research Articles

A nonlinear fractional fishery resource system model with Crowley–Martin functional response under Mittag-Leffler kernel

This article studies the action of predators and the predator-dependent functional response in the fishery resource model. We employ the Atangana–Baleanu–Caputo fractional derivative to study the proposed fractional fishery resource model in the presence of predators with Crowley–Martin functional response. We present a theoretical and numerical analysis of the governing nonlinear differential equations of the model consisting of the biomass density of the fish population inside the unrestricted fishing zone, the biomass density of the fish population inside the reserve or restricted fishing zone, and the predator population. Using the fixed point theory and nonlinear analysis, we establish the existence and uniqueness results of the proposed ABC fractional fishery resource model. We establish stability analysis of the fishery model using the Ulam–Hyers stability approach. The numerical scheme of the fractional Adams–Bashforth method is provided and the approximate solutions for the model under consideration are given and discussed. We observe that an increase in the fish capturing rates increases the size of the predator population and reduces the fish subpopulations. To maintain a high number of fish species, we recommend a control measure to reduce the fish capturing rate by the predators.

Complex dynamics in a two species system with Crowley–Martin response function: Role of cooperation, additional food and seasonal perturbations

This research article investigates the interaction between prey and a generalist predator, considering the effect of hunting cooperation. The predator–prey interaction is modeled using a predator dependent functional response, specifically the Crowley–Martin type. System’s dynamics are explored using both analytical and numerical techniques. Feasible equilibria are analyzed, and their local stability is determined. Various bifurcations in the system are explored, and one-and two-parameter bifurcation structures are constructed to unveil complex dynamical behaviors. Our findings reveal that both prey and predator face extinction when the predator growth rate from alternative food sources exceeds certain threshold values. However, prey extinction is driven by the higher levels of hunting cooperation among predators, while the availability of external food sources enhances the system’s stability and persistence. Furthermore, we enhance the model by introducing seasonal perturbations, acknowledging the influence of seasonal variations on ecological parameters. The system exhibits diverse patterns in response to seasonality, contributing to a deeper understanding of the dynamics in predator–prey interactions.

Understanding the hydra effect in predator-dependent functional response models

Understanding the hydra effect in predator-dependent functional response models

Dynamical Analysis of a Beddington–DeAngelis Interacting Species System with Prey Harvesting

A reaction–diffusion interacting species system with Beddington–DeAngelis functional response that has been proposed in the environment of mathematical ecology, which provides the rise to spatial pattern formation, is investigated and associated with the models of deterministic dynamics. The dynamical behaviour of a generalist predator–prey system with linear harvesting of each species and predator-dependent functional response is fully analyzed. Conditions of stability behaviour of the interior equilibrium point are established properly. Furthermore, we have recognized that the unique positive equilibrium point of the system is globally stable via appropriate Lyapunov function structure, which signifies that appropriate harvesting has no impact on the persistence property of the harvesting system. Also, we establish the conditions for the existence of bifurcation phenomena including a saddle-node bifurcation and a Hopf bifurcation. Subsequently, complete analysis regarding the impact of harvesting is carried out, and an interesting decision is that under some appropriate constraints, harvesting has immense impact on the final size of the interacting species. In addition, in accordance with Turing’s ideas on morphogenesis , our analysis shows that harvesting effort in a reaction–diffusion predator–prey system plays a vital function for geological conservation of interacting species. Finally, we discuss sufficient conditions for the existence of bionomic equilibrium point and the optimal harvesting policy attained by using the Pontryagin maximal principle.

Stability and Bifurcation Analysis of Hassell–Varley Prey–Predator System with Fear Effect

In this work, we study a prey–predator system with Hassell–Varley functional response, which is a predator-dependent functional response. Our goal is to study the dynamics of the system in the presence of fear of predation risk. Hassell–Varley functional response is used when the predator population forms a fixed number of tight groups like for foraging purposes. Those predators can be terrestrial and aquatic. We assume that the birth rate of prey population reduces due to fear of the predator. Here, we have discussed the existence of biologically relevant equilibria and derived the condition for local asymptotic stability. Further, permanence and persistence are discussed to understand the dynamics of the system. The system undergoes Hopf bifurcation for both: the fear parameter f and interference coefficient $$\gamma $$ . Stability and the direction of Hopf bifurcation are also discussed. We observe that limit cycle oscillation can be prevented by increasing the cost of fear of the predator population. Numerical simulation is performed to substantiate our analytical findings.

Impact of generalist type sexually reproductive top predator interference on the dynamics of a food chain model

Food uptake ability of higher trophic level species are more complicated and interesting due to their choice and availability of food and consequently their growth. In the present study, the effect of top predator interference on the dynamics of a tritrophic food chain model involving an intermediate and a generalist type top predator are considered. It is assumed that the interaction between the logistically growing prey and intermediate predator follows the Volterra scheme, while that between the top predator and its favorite food i.e. the intermediate predator follows Crowley–Martin type functional response. Also, it is considered that the growth of top predator is by sexual reproduction. The boundedness of the system, existence of an attracting set, local and global stability of non-negative equilibrium points and uniform persistence are established by very complicated and highly nonlinear biological considerations. With the help of empirical results from different field and laboratory experiment, it is clarified that Crowley–Martin type functional response gives better result than other predator dependent functional responses for the issue of interference among own population. Also it is clarified from literature survey that high trophic species are mostly generalist type i.e. they can survive rather than their favorite food depending upon the present environmental situation and their growth rate depends upon the mating success. Moreover, increasing the top predator interference stabilizes the system, while increasing the normalization of the residual reduction in the top predator population destabilizes the system. Different numerical examples support these ecological phenomenon.

Traveling Wave Solutions for Some Classes of Diffusive Predator–Prey Models

We investigate the traveling wave solutions for some types of diffusive Predator–prey systems, including the models with prey and predator dependent functional response, that have served as models to study the dynamics of Predator–prey interaction. The method used to show the existence of traveling waves and to identify minimum wave speed consists of two steps. First we obtain globally defined solutions by a shooting argument that is a modification of a recently developed method. We then show the convergence of these global solutions to an interior equilibrium point by the construction of a Liapunov function.

Predator‐dependent functional response in wolves: from food limitation to surplus killing

The functional response of a predator describes the change in per capita kill rate to changes in prey density. This response can be influenced by predator densities, giving a predator-dependent functional response. In social carnivores which defend a territory, kill rates also depend on the individual energetic requirements of group members and their contribution to the kill rate. This study aims to provide empirical data for the functional response of wolves Canis lupus to the highly managed moose Alces alces population in Scandinavia. We explored prey and predator dependence, and how the functional response relates to the energetic requirements of wolf packs. Winter kill rates of GPS-collared wolves and densities of cervids were estimated for a total of 22 study periods in 15 wolf territories. The adult wolves were identified as the individuals responsible for providing kills to the wolf pack, while pups could be described as inept hunters. The predator-dependent, asymptotic functional response models (i.e. Hassell-Varley type II and Crowley-Martin) performed best among a set of 23 competing linear, asymptotic and sigmoid models. Small wolf packs acquired >3 times as much moose biomass as required to sustain their field metabolic rate (FMR), even at relatively low moose abundances. Large packs (6-9 wolves) acquired less biomass than required in territories with low moose abundance. We suggest the surplus killing by small packs is a result of an optimal foraging strategy to consume only the most nutritious parts of easy accessible prey while avoiding the risk of being detected by humans. Food limitation may have a stabilizing effect on pack size in wolves, as supported by the observed negative relationship between body weight of pups and pack size.

Prey–predator dynamics in rotifers: density‐dependent consequences of spatial heterogeneity due to surface attachment

Classical models of prey-predator interactions assume that per capita prey consumption is dependent on prey density alone and that prey consumption (functional response) and consumer proliferation (numerical response) operate on the same timescales and without time lags. Several modifications have been proposed for resolving this timescale discrepancy, including variants where the functional response depends on both prey and predator densities. A microcosm system with the rotifer Brachionus 'Nevada' feeding on the prasinophyte Tetraselmis sp. showed significant (P < 0.0005) increases in steady-state biomasses of both prey and predators with increasing carrying capacity (represented by total phosphorus of the growth medium), which is inconsistent with predictions based on the traditional prey-only-dependent functional response. We provide data indicating that surfaces where the predator can attach provide a high-quality habitat for rotifers, which can result in a predator-dependent functional response. We also show that partitioning between the attached and free-swimming habitats was fast compared to the timescale of the numerical response. When attached to surfaces, rotifers maximized net energy gain by avoiding the high cost of swimming and by increased food capture due to reduced viscous drag. A mathematical model with prey-dependent functional response and wall-attached and free-swimming fractions of the population describes our data adequately. We discuss the implications of this finding for extrapolating microcosm experiments to systems with other surface-to-volume ratios, and to what extent our findings may apply to other popular model organisms for prey-predator interaction.

A note on the existence of periodic solutions in discrete predator–prey models

This note investigates the existence of periodic solutions for discrete semi-ratio-dependent predator–prey models with functional response. New criteria are derived in which this system with prey-dependent (monotonic or non-monotonic) and predator-dependent functional response bounded by polynomials in R + always has at least one ω -periodic solution, which improves and extends many previous results [1–5]. Some interesting numerical examples are given to illustrate our results.

Dynamics of a three species food chain model with Crowley–Martin type functional response

In this paper, a three species food chain model, consisting of a hybrid type of prey-dependent and predator-dependent functional responses, is investigated analytically as well as numerically. The local and global stability analysis is carried out. The persistence conditions are established. Bifurcation diagrams are obtained for biologically feasible parameters. The results show that the system exhibits rich complexity features such as stable, periodic and chaotic dynamics.

Periodic solutions for impulsive semi-ratio-dependent predator–prey systems

This paper investigates the existence of periodic solutions for an impulsive semi-ratio-dependent predator–prey system with most popular prey-dependent (monotone or non-monotone) and predator-dependent functional response. Sharp sufficient conditions are derived by the invariance property of homotopy and analysis technique. The presented criteria improve and extend many previous results in the literature.

FUNCTIONAL RESPONSES: A QUESTION OF ALTERNATIVE PREY AND PREDATOR DENSITY

Throughout the study of ecology, there has been a growing realization that indirect effects among species cause complexity in food webs. Understanding and predicting the behavior of ecosystems consequently depends on our ability to identify indirect effects and their mechanisms. The present study experimentally investigates indirect interactions arising between two prey species that share a common predator. In a natural field experiment, we introduced different densities of mealworms (Tenebrio molitor), an alternative prey, to a previously studied predator-prey system in which paper wasps (Polistes dominulus) preyed on shield beetle larvae (Cassida rubiginosa). We tested if alternative prey affects predation on the first prey (i.e., the predator-dependent functional response of paper wasps) by modifying either interference among predators or the effective number of predators foraging on shield beetles. Presence of mealworms significantly reduced the effective number of predators, whereas predator interference was not affected. In this way, the experimentally introduced alternative prey altered the wasps' functional response and thereby indirectly influenced C. rubiginosa density. In all prey-density combinations offered, paper wasps constantly preferred T. molitor. This led to an asymmetrical, indirect interaction between both prey species: an increase in mealworm density significantly relaxed predation on C. rubiginosa, whereas an increase in C. rubiginosa density intensified predation on mealworms. Such asymmetrical outcomes of a fixed food preference can significantly affect the population dynamics of the species involved. In spite of the repeated finding of a Type III functional response in this system, our experiment did not reveal switching behavior in paper wasps. The variety of mechanisms underlying direct and indirect interactions within our study system exemplifies the importance of incorporating alternative prey when investigating the impact of a generalist predator on a focal prey population under realistic field conditions.

Emergence of ratio-dependent and predator-dependent functional responses for pollination mutualism and seed parasitism

Prey ( N) dependence [ g( N)], predator ( P) dependence [ g( P) or g( N, P)], and ratio dependence [ f( P/ N)] are often seen as contrasting forms of the predator's functional response describing predator consumption rates on prey resources in predator–prey and parasitoid–host interactions. Analogously, prey-, predator-, and ratio-dependent functional responses are apparently alternative functional responses for other types of consumer–resource interactions. These include, for example, the fraction of flowers pollinated or seeds parasitized in pollination (pre-dispersal) seed-parasitism mutualisms, such as those between fig wasps and fig trees or yucca moths and yucca plants. Here we examine the appropriate functional responses for how the fraction of flowers pollinated and seeds parasitized vary with the density of pollinators (predator dependence) or the ratio of pollinator and flower densities (ratio dependence). We show that both types of functional responses can emerge from minor, but biologically important variations on a single model. An individual-based model was first used to describe plant–pollinator interactions. Conditional upon on whether the number of flowers visited by the pollinator was limited by factors other than search time (e.g., by the number of eggs it had to lay, if it was also a seed parasite), and on whether the pollinator could directly find flowers on a plant, or had to search, the simulation results lead to either a predator-dependent or a ratio-dependent functional response. An analytic model was then used to show mathematically how these two cases can arise.

Predator-dependent functional responses and interaction strengths in a natural food web

Predator-dependent functional responses decouple predation mortality from fluctuations in predator abundance and therefore can prevent strong "top-down" interaction strengths in food webs. We evaluated whether contrasts in the functional response of Baltic Sea cod (Gadus morhua) were consistent with the contrasting population dynamics of two prey species, herring (Clupea harengus) and sprat (Sprattus sprattus): sprat abundance increased nearly threefold following a sharp decline in the cod population (a strong interaction), whereas herring abundance failed to increase (a weak interaction). We found striking differences in the functional response of cod on alternative prey, and these were consistent with the observed patterns in interaction strengths. Cod predation was the dominant source of mortality for age-1 and age-2 sprat but was only important for age-1 herring. Moreover, the magnitude of predation mortality on age-1 and age-2 sprat was highly sensitive to cod biomass, whereas predation mortality on herring was only moderately sensitive to cod biomass. These analyses suggest the possibility that food webs are comprised of linkages that vary with respect to the magnitude and importance of predation mortality and how this mortality varies with changes in predator abundance.

Does mutual interference always stabilize predator–prey dynamics? A comparison of models

Based on a qualitative analysis of ODE systems, the dynamic properties of alternative predator–prey models with predator-dependent functional response have been compared in order to study the role that predator interference plays in the stabilisation of trophic systems. The models considered for interference have different mathematical expressions and different conceptual foundations. Despite these differences, they give essentially the same qualitative results: when interference is low, increasing it has a positive effect on asymptotic stability and thus on the resilience of the biological system. When it is high, it is the contrary (with logistic prey growth, increasing the interference parameter ensures stability but leads to very small predator densities). Possible consequences on the evolution of the interference level in real ecosystems are discussed. To cite this article: R. Arditi et al., C. R. Biologies 327 (2004).

Seasonally perturbed prey–predator system with predator-dependent functional response

The effect of seasonality on the prey–predator model with predator-dependent trophic function is investigated analytically as well as numerically. The effect of periodic variations is considered on two different parameters of the system: the growth rate of prey and the death rate of the predators. The two parameters may not be in the same phase. The behavior of the system is simulated and bifurcation diagrams are obtained for different parameters. The results show that seasonality in two different parameters with or without phase difference can give rise to multiple attractors, including chaos, with variations in critical parameters.

Functional Responses with Predator Interference: Viable Alternatives to the Holling Type II Model

A predator's per capita feeding rate on prey, or its functional response, provides a foundation for predator–prey theory. Since 1959, Holling's prey-dependent Type II functional response, a model that is a function of prey abundance only, has served as the basis for a large literature on predator–prey theory. We present statistical evidence from 19 predator–prey systems that three predator-dependent functional responses (Beddington-DeAngelis, Crowley-Martin, and Hassell-Varley), i.e., models that are functions of both prey and predator abundance because of predator interference, can provide better descriptions of predator feeding over a range of predator–prey abundances. No single functional response best describes all of the data sets. Given these functional forms, we suggest use of the Beddington-DeAngelis or Hassell-Varley model when predator feeding rate becomes independent of predator density at high prey density and use of the Crowley-Martin model when predator feeding rate is decreased by higher predator density even when prey density is high.

FUNCTIONAL RESPONSES WITH PREDATOR INTERFERENCE: VIABLE ALTERNATIVES TO THE HOLLING TYPE II MODEL

A predator's per capita feeding rate on prey, or its functional response, provides a foundation for predator–prey theory. Since 1959, Holling's prey-dependent Type II functional response, a model that is a function of prey abundance only, has served as the basis for a large literature on predator–prey theory. We present statistical evidence from 19 predator–prey systems that three predator-dependent functional responses (Beddington-DeAngelis, Crowley-Martin, and Hassell-Varley), i.e., models that are functions of both prey and predator abundance because of predator interference, can provide better descriptions of predator feeding over a range of predator–prey abundances. No single functional response best describes all of the data sets. Given these functional forms, we suggest use of the Beddington-DeAngelis or Hassell-Varley model when predator feeding rate becomes independent of predator density at high prey density and use of the Crowley-Martin model when predator feeding rate is decreased by higher predator density even when prey density is high.

A Formal Derivation of the “Beddington” Functional Response

In ecological modeling the interaction between a predator and its prey, is usually implemented as a linear or saturated function of the prey density. The main advantages of such a “functional response” are its simplicity, general applicability, and well understood mechanistic basis. In systems where predators compete directly for the available prey however, the functional response should depend not only on the prey density but also on the predator density. We aim here to devise a simple and generic “predator-dependent” functional response. We derive such a functional response by making quasi-steady-state assumptions for models, in which we allow predators and prey to form interaction complexes. We end up with the—previously proposed—“Beddington” functional response. Because of our formal derivation this simple predator-dependent functional response is now based on clear mechanistic reasoning. The direct predator interference of this functional response emerges from the interaction between a predator and a prey, and not from direct predator-predator interactions. We conclude however, that, although the Beddington functional response is generic for a two-dimensional system of one prey interacting with one predator, it is difficult to generalize it to higher dimensional systems consisting of several prey and predator species.