Growth Rate Of Prey Research Articles

Hopf bifurcation and stability analysis of the Rosenzweig-MacArthur predator-prey model with stage-structure in prey.

We consider a stage-structure Rosenzweig-MacArthur model describing the predator-prey interaction. Here, the prey population is divided into two sub-populations namely immature prey and mature prey. We assume that predator only consumes immature prey, where the predation follows the Holling type II functional response. We perform dynamical analysis including existence and uniqueness, the positivity and the boundedness of the solutions of the proposed model, as well as the existence and the local stability of equilibrium points. It is shown that the model has three equilibrium points. Our analysis shows that the predator extinction equilibrium exists if the intrinsic growth rate of immature prey is greater than the death rate of mature prey. Furthermore, if the predation rate is larger than the death rate of predator, then the coexistence equilibrium exists. It means that the predation process on the prey determines the growing effects of the predator population. Furthermore, we also show the existence of forward and Hopf bifurcations. The dynamics of our system are confirmed by our numerical simulations.

Analysis of a plankton–fish model with external toxicity and nonlinear harvesting

In this paper, we consider a three species plankton–fish system that incorporates external toxicity and nonlinear harvesting. We consider that the growth of species are affected directly or indirectly by an external toxic substance and the feeding of the predator on the affected prey is considered as Holling type II functional response. All the possible biological feasible equilibrium points are determined analytically as well as numerically and performed stability analysis around these equilibrium points. It is shown that the system undergoes for Hopf bifurcation when the growth rate of prey passes some threshold value. Furthermore, Pontryagin’s maximum principle has been applied to obtain optimal control of harvesting to maximize the benefit as well as the conservation of the ecosystem. We perform numerical simulations to justify and illustrate our analytical results. Some numerical tools such as phase portraits, time evaluation and bifurcation diagrams are presented to ensure the complex dynamics in the system. Period doubling cascade route to chaos is examined and validated through the numerical calculation of Lyapunov exponents and sensitivity analysis.

Synchronization and Bursting Activity in the Model for Two Predator-Prey Systems Coupled By Predator Migration

The papers is devoted to a model for two non-identical predator-prey communities coupled by migration and characterized by logistic growth of prey and Holling type II functional response. The coupling is a predator migration at constant weak rate. The non-identity is the difference in the prey growth rates or predator mortalities in each patch. We studied the equilibrium states of model and scenarios of loss of their stability and emerge of complex periodic solutions. It was shown that in some domains of the parameter space there is a bursting activity which are that the dynamics of two communities contain segments of slowly resting dynamic (as part of a fast-slow cycle or canard) and regular bursts of spikes. In the resting part, the dynamics of the second community, as a rule, follow the slow changes in the first community, i.e. the dynamics in different patches are synchronous. But in the fast part there is only phase synchronization between the fast-slow cycle in first patch and bursts in second. We classified the scenarios of transition between different types of bursting activity by location spiking manifold and canard. These types differ not so much in size, shape or numbers of spikes as in the order of bursts emerge relative a slow-fast cycle. In a typical case the start of burst (divergent fast oscillations) coincides with the minimum numbers or quasi-extinction of prey in the first patch. After a rapid increase in the prey number in the first patch, diverging fluctuations give way to damped in the second patch. Such dynamics correspond to the rhombus-wave shape of spikes cluster. Another case is interesting, when the burst of spikes is formed after the full recovery of prey and with a certain predator number in the first patch. In this case, the spikes cluster takes the shape of a triangle-wave or a truncated rhombus-wave. It was shown that transitions between these types of bursts are accompanied by a change in the oscillation period and the degree of synchronization. Triangular-wave bursters correspond to complete synchronization of the predator dynamics in the resting part and rhomboid-wave correspond to antiphase synchronization. In the fast part with many spikes, communities are completely asynchronous to each other.

Fear factor in a prey–predator system in deterministic and stochastic environment

In this article, a prey–predator model has been studied where apart from direct predation the prey population is affected by the fear induced from predators. The basic reproduction of the prey population is reduced as a cost of fear. Biologically well posedness of the model system has been shown through positivity and boundedness of solutions. Existence criterion and stability analysis of the non-negative equilibrium points have been discussed and necessary conditions for uniform persistence have been derived for corresponding non-autonomous system. Also, sufficient conditions for the existence of positive periodic solution of the non-autonomous system have been established by utilizing the coincidence degree theorem. We further show that the positive periodic solution is global attractor under certain conditions. Next, we construct the stochastic model based upon the deterministic setup by perturbing the intrinsic growth rate of prey and natural mortality rate of predators. It has been shown that the stochastic system admits unique positive global solution initiates from anywhere in the interior of the positive quadrant. The sufficient conditions for extinction, non-persistence and weakly persistence of both the species have been derived along with the stochastic permanence of the system. We have verified our analytical results and make a comparison between the deterministic and stochastic setup by exhaustive numerical simulations.

Density Dependent Delayed Migration for Rosenzweig-Macaurther Model with Holling Type II Predator Functional Response

The model describing the interaction between the predator and prey species is referred to as a predator-prey model. The migration of these species from one patch to another may not be instantaneous. This may be due to barriers such as a swollen river or a busy infrastructure through the natural habitat. Recent predator-prey models have either incorporated a logistic growth for the prey population or a time delay in migration of the two species. Predator-prey models with logistic growth that integrate time delays in density-dependent migration of both species have been given little attention. A Rosenzweig-MacAurther model with density-dependent migration and time delay in the migration of both species is developed and analyzed in this study. The Analysis of the model when the prey migration rate is greater than or equal to the prey growth rate, the two species will coexist, otherwise, at least one species will become extinct. A longer delay slows down the rate at which the predator and prey population increase or decrease, thus aecting the population density of these species. The prey migration due to the predator density does not greatly affect the prey density and existence compared to the other factors that cause the prey to migrate. These factors include human activities in the natural habitats like logging and natural causes like bad climatic conditions, limited food resources and overpopulation of the prey species in a patch among others.

Long-time behavior of a diffusive prey–predator system with Beddington–DeAngelis functional response in heterogeneous environment

In this paper, we study a diffusive prey–predator model with Beddington–DeAngelis functional response when the intraspecific crowding effect for the prey population disappears in some subdomain $$\Omega _0$$ of their whole habitat. We are concerned about the global dynamics of the system and discuss it based on two coefficients: the growth rate of prey $$\lambda $$ and that of predator $$\mu $$. In particular, the results show that, with the degeneracy of prey population’s intraspecific crowding effect in the subdomain $$\Omega _0$$, the density of prey may tend to infinity in $$\Omega _0$$. On the other hand, the unboundedness of prey means that the solutions of the system lose compactness which may bring difficulty to investigate the long-time behavior of the solutions. Finally, some numerical simulations are presented to support and strengthen our theoretical analysis.

A prey–predator model with refuge for prey and additional food for predator in a fluctuating environment

Refuge for prey and additional food for predator together with environmental fluctuation is a very interesting topic for dynamical system. In this work we have studied a prey–predator model in presence of some additional food to predators and incorporating refuge for prey which protects certain amount of prey. In this model prey grows logistically in absence of predator. Functional response of predator towards additional food and prey is derived considering mutual interference among the individuals of the predator. We have introduced Gaussian white noise to model the influence of environmental fluctuations on the growth rate of prey and the death rate of predator. We have shown, how Gaussian white noise plays an important role for extinction and persistence of the system. Some important properties such as positivity, boundedness, uniform continuity etc. of the solution are proved. In extinction scenario we have found that extinction of prey population propel predator population to extinct. It is considered that prey refuge depends on biomass of prey and predator population both. In numerical simulation we have justified how this consideration is more actual to the reality. Also it is shown the importance of additional food to predator as prey may not be available (prey may get refuge) for predation.

Effects of Additional Food on the Dynamics of a Three Species Food Chain Model Incorporating Refuge and Harvesting

AbstractIn this paper, a three species food chain model has been developed among the interaction of prey, predator and super predator. It is assumed that the predator shows refuge behavior to the super predator. It is also assumed that a certain amount of additional food will be supplied to the super predator. It is considered that the predator population is benefiting partially from the additional food. To get optimal harvesting of super predator the Pontryagin’s maximum principle has been used. It is found that super predator may be extinct if harvesting rate increase. It is observed that as the refuge rate increases, predator population gradually increases, but super predator population decreases. Also, it is found that our proposed system undergoes oscillatory or periodic behavior as the value of refuge rate (m1), harvesting rate (E), the intrinsic growth rate of prey (r), carrying capacity of prey (k) and conservation rate of prey (c1) varies for some certain range of these parameters. It is found that this study may be useful for the increase of harvesting of a super predator by supplying the additional food to our proposed system.

Dynamics of a predator-prey system with fear and group defense

We study the dynamics of a prey-predator interaction model that incorporates: (1) reduction of prey growth rate, in the form of fear effect, in presence of predator; and (2) group defense of prey, against predation, by using the Monod-Haldane type functional response. Moreover, we interrelate these two factors, through the predator-taxis sensitivity, as the total time or energy for foraging and defense is constant for prey. If the prey invests more time or energy for group defense, then reproduction may decrease due to that investment. We provide detailed mathematical results, including, basic dynamical properties, existence of positive equilibria, asymptotic stability of all equilibria, Hopf-bifurcation, direction and stability of bifurcated periodic solutions. We also provide some global features and possible occurrence of multi-stability in our model. Furthermore, we perform detailed numerical simulations to validate our mathematical results numerically. Our mathematical and numerical results suggest that the predator-taxis sensitivity should be less than some threshold density, for possible survivability of predator. We provide some sensitivity analysis of our model solutions with respect to the three important model parameters, namely, the predator-taxis sensitivity, level of fear, and the tolerance limit of predator. We can observe that the perturbation of the tolerance limit of predator has the greatest influence over model dynamics. Initially, the predator-taxis sensitivity has a positive effect on prey as its decreases the killing rate, however, for long run, its effect is negative on both the solutions, as it decreases the growth rate of prey, which affects overall fitness of both the populations. Our results may provide some useful biological insights on predator-prey interactions.

Rosenzweig-Macaurther Model with Holling type II Predator Functional Response for Constant Delayed Migration

Predator-prey models describe the interaction between two species, the prey which serves as a food source to the predator. The migration of the prey for safety reasons after a predator attack and the predator in search of food, from a patch to another may not be instantaneous. In this paper, a Rosenzweig-MacAurther model with a Holling-type II predator functional response and time delay in the migration of both species is developed and analysed. Stability analysis of the system shows that depending on the prey growth and prey migration rates either both species go to extinction or co-exist. Numerical simulations show that a longer delay in the migration of the species leads makes the model to stabilize at a slower rate compared to when the delay is shorter. Relevant agencies likethe Kenya Wildlife Service should address factors that slow down migration of species, for example, destruction of natural habitats for human settlement and activities, which may cause delay in migration.

The effect of the fear factor on the dynamics of a predator-prey model incorporating the prey refuge.

In this paper, we investigate the dynamics of an improved Leslie-Gower predator-prey model which is characterized by the reduction of the prey growth rate due to fear of the predator (i.e., antipredator behavior). The value of this study lies in two aspects: mathematically, (i) it provides the existence and the stability of the positive equilibrium; (ii) it gives the existence of the Hopf bifurcation and limit cycle; and (iii) it shows the mechanisms of the fear factor and the prey refuge on the level of the positive equilibrium. Biologically, we find that the influence of the fear factor is complex: (i) increasing the level of fear can cause the level of the population density to decrease and the prey to become extinct; (ii) the effect of the cost of fear on the stability of the positive equilibrium is rich and complex: it can either destabilize the stability and benefit the emergency of the periodic behavior or stabilize the system by excluding the existence of periodic solutions; (iii) with a fixed level of fear, the prey refuge is beneficial to the coexistence of the prey and the predator, and with the increase of the level of the prey refuge, the positive equilibrium may change from stable spiral sink to unstable spiral source to stable spiral sink. These results may enrich the dynamics of the predator-prey systems.

Comparative studies on a predator–prey model subjected to fear and Allee effect with type I and type II foraging

In ecology, the interactions between prey and predator can be direct or indirect. Predators can affect prey population by means of direct consumption and by inducing a non-consumptive fear effect in the prey population. In our present study, we investigate the impact of an indirect prey–predator interaction by considering the effect of fear of predation coupled with predation-driven Allee effect on the prey population. We propose and analyze two-species prey–predator models where the growth rate of prey is affected by a combined non-consumptive Allee-cum–fear effect induced by the predator and due to two different foraging behaviour of the predator. We investigate and compare the dynamical behaviour of the systems with Holling type-I and Holling type-II foraging rates of the predator. We observe that the system with type-II foraging rate of the predator has a wider basin of attraction of the stable coexistence equilibrium compared to the system with type-I foraging rate. We also observe that due to the gradual increase of fear level, the systems exhibit oscillatory dynamics followed by multiple stability switching.

Long-term aquaria study suggests species-specific responses of two cold-water corals to macro-and microplastics exposure

Plastic pollution has been identified as a major threat for coastal marine life and ecosystems. Here, we test if the feeding behaviour and growth rate of the two most common cold-water coral species, Lophelia pertusa and Madrepora oculata, are affected by micro- or macroplastic exposures. Low-density polyethylene microplastics impair prey capture and growth rates of L. pertusa after five months of exposure. Macroplastic films, mimicking plastic bags trapped on deep-sea reefs, had however a limited impact on L. pertusa growth. This was due to an avoidance behaviour illustrated by the formation of skeletal ‘caps’ that changed the polyp orientation and allowed its access to food supply. On the contrary, M. oculata growth and feeding were not affected by plastic exposure. Such a species-specific response has the potential to induce a severe change in coral community composition and the associated biodiversity in deep-sea environments.

Uniqueness and stability of positive solutions for a diffusive predator–prey model in heterogeneous environment

In this paper, we study a spatially heterogeneous predator–prey model where the interaction is governed by a Crowley-Martin type functional response. It is shown that the predator competition in this model plays an extremely important role. A good understanding of the existence, uniqueness and stability of positive solutions is obtained when the predator competition is strong. In particular, we show that when the predator competition is strong, the model has at most one positive steady-state solution for any $$\mu \in {{\mathbb {R}}}$$ , and it is globally asymptotically stable for any $$\mu >0$$ (if it exists). Additionally, we show that when the prey growth rate is over a critical value, the two populations stabilize at a unique coexistence state if the predator growth rate is strong.

Allee effects under climate change

Understanding how climate change affects population dynamics is crucial for assessing future of biodiversity. Here I ask how can Allee effects, occurring when mean individual fitness is reduced in rare populations, respond to increasing temperature. Despite the role Allee effects play in ecology of invasive, threatened and harvested populations, impacts of climate change on Allee effects are practically unknown. Analysis of two population models reveals that whereas the Allee effect driven by predation generally weakens as temperature increases, the Allee effect due to need of finding mates is predicted to become stronger when warming occurs. For the former model, the metabolic theory suggests that with increasing temperature prey growth rate should increase faster than predator attack rate. Increasing temperature thus weakens the Allee effect. In the latter, gypsy moth population model, mating rate increases with warming due to enhanced female−male encounter rate and temperature‐induced modifications in female and male adult emergence distributions. However, male and female mortality rates increase, too and the net effect is strengthening of the Allee effect. These results have repercussions also for pest control, indicating that augmentation of biocontrol agents may perhaps be not as effective as using pesticides or disrupting mating.

Stability and bifurcation analysis of a discrete predator\u2013prey system with modified Holling\u2013Tanner functional response

In this paper, we study a discrete predator–prey system with modified Holling–Tanner functional response. We derive conditions of existence for flip bifurcations and Hopf bifurcations by using the center manifold theorem and bifurcation theory. Numerical simulations including bifurcation diagrams, maximum Lyapunov exponents, and phase portraits not only illustrate the correctness of theoretical analysis, but also exhibit complex dynamical behaviors and biological phenomena. This suggests that the small integral step size can stabilize the system into the locally stable coexistence. However, the large integral step size may destabilize the system producing far richer dynamics. This also implies that when the intrinsic growth rate of prey is high, the model has bifurcation structures somewhat similar to the classic logistic one.

Semi Non-Standard Trimean Algorithm for Rosenzweig-MacArthur Interaction Model

Most real world natural systems are shows seasonal behaviour due to seasonal environmental or climate change. As a results, many species display seasonal changes in their life history parameters. It is crucial to comprehend how the seasonal forcing controls the behaviour of the population dynamics. The Rosenzweig-MacArthur model is a system with at least two ordinary differential equations used in population dynamics to model the interaction of predator and prey bonding. Rosenzweig-MacArthur model overcome the weakness of Lotka-Volterra model to simulate interaction between two species. In Rosenzweig-MacArthur model, logistic growth rate of prey is resource limited. The model utilizes Holling type II as the functional response representation. The purpose of the study is to construct method to improve simulation on the behaviour of interactions between species and predicting equilibrium point accurately and fast. Current methods seem to predict accurately the equilibrium point if only small mesh size used. Using small mesh size will require long simulation time to predict the equilibrium point. Able to increase the mesh size will increase the speed of predicting the equilibrium point. In this paper, we propose three new semi non-standard trimean algorithms to simulate the behaviour of interaction between species represented by Rosenzweig-MacArthur model. The new algorithms apply a hybrid of semi non-standard approach and trimean to approximate the nonlinear terms in the differential equation model. Two cases of experiment conducted to examine the performance of all three semi non-standard schemes. Result shows that all three new semi non-standards schemes accurately predict the equilibrium point (0.25, 0.46875) even using big mesh size ( and ) for both cases. Thus, all three semi non-standard schemes fulfil the purpose of this study.

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.

Stochastic prey-predator system with foraging arena scheme

• The stochastic foraging arena prey-predator model is studied. • The stochastic system is obtained by introducing two independent Brownian motions. • The long-time behaviours and the stationary distribution are investigated. In this paper, we extend the foraging arena model describing the dynamics of prey-predator abundance from a deterministic framework to a stochastic one. This is achieved by introducing the environmental noises into the growth rate of prey as well as the death rate of predator populations. We then prove that this stochastic differential equation (SDE) has a unique global positive solution. The long-time behaviours of the system are then developed. Furthermore the existence of a stationary distribution is pointed out under certain parametric restrictions. All the results are illustrated by the computer simulations.

Modeled larval fish prey fields and growth rates help predict recruitment success of cod and anchovy in the North Sea

Modeled larval fish prey fields and growth rates help predict recruitment success of cod and anchovy in the North Sea