scholarly journals Effect of fear on two predator-one prey model in deterministic and fluctuating environment

2021 ◽  
pp. 1-17
Author(s):  
Debasis Mukherjee
Keyword(s):  
Deterministic Model ◽  
Antipredator Behavior ◽  
Point Of View ◽  
Uniform Persistence ◽  
Reproduction Rate ◽  
Multiple Predators ◽  
Predator Prey ◽  
Predator Attack ◽  
Global Positive Solution ◽  
Independent Effects

Recent ecological studies on predator-prey interactions has concentrated on determining the impacts of antipredator behavior due to fear of predators. These studies are mainly confined into one predator-one prey system. But in case of multiple predator attack on single prey species, fear mechanism is still unknown. The combined impact of multiple predator often cannot be anticipated from their independent effects. So coexistence of multiple predators and prey’s fitness becomes an important issue from an ecological point of view. Based on the above observations, we proposed and analyzed a model consisting of two competing predator sharing a common prey where prey’s reproduction rate is affected due to fear generated by the predators. We first study the boundedness, uniform persistence, stability and Hopf bifurcation of the deterministic model. Thereafter, we have investigated the existence and uniqueness of the global positive solution, boundedness, asymptotic stability of the stochastic model.  Numerical examples are provided to support our obtained  results.

scholarly journals Stochastic dynamics of consumer-resource interactions

2021 ◽  
Author(s):  
Abhyudai Singh
Keyword(s):  
Attack Rate ◽  
Stochastic Dynamics ◽  
Reproduction Rate ◽  
Population Densities ◽  
Predator Prey ◽  
Density Dependent ◽  
Predator Prey Model ◽  
Predator Attack ◽  
Resource Interactions ◽  
Consumer Resource

AbstractThe interaction between a consumer (such as, a predator or a parasitoid) and a resource (such as, a prey or a host) forms an integral motif in ecological food webs, and has been modeled since the early 20th century starting from the seminal work of Lotka and Volterra. While the Lotka-Volterra predator-prey model predicts a neutrally stable equilibrium with oscillating population densities, a density-dependent predator attack rate is known to stabilize the equilibrium. Here, we consider a stochastic formulation of the Lotka-Volterra model where the prey’s reproduction rate is a random process, and the predator’s attack rate depends on both the prey and predator population densities. Analysis shows that increasing the sensitivity of the attack rate to the prey density attenuates the magnitude of stochastic fluctuations in the population densities. In contrast, these fluctuations vary non-monotonically with the sensitivity of the attack rate to the predator density with an optimal level of sensitivity minimizing the magnitude of fluctuations. Interestingly, our systematic study of the predator-prey correlations reveals distinct signatures depending on the form of the density-dependent attack rate. In summary, stochastic dynamics of nonlinear Lotka-Volterra models can be harnessed to infer density-dependent mechanisms regulating consumer-resource interactions. Moreover, these mechanisms can have contrasting consequences on population fluctuations, with predator-dependent attack rates amplifying stochasticity, while prey-dependent attack rates countering to buffer fluctuations.

Dynamics of delayed stochastic predator–prey models with feedback controls based on discrete observations

2017 ◽  
Vol 10 (03) ◽  
pp. 1750040 ◽  
Author(s):  
Dianli Zhao ◽  
Sanling Yuan
Keyword(s):  
Numerical Simulations ◽  
Stochastic System ◽  
Necessary Conditions ◽  
Deterministic Model ◽  
Sufficient Conditions ◽  
Discrete Observations ◽  
Feedback Controls ◽  
Predator Prey ◽  
Global Positive Solution ◽  
Predator Prey Models

In this paper, we concern a class of the generalized delayed stochastic predator–prey models with feedback controls based on discrete observations. The existence of global positive solution is given first. Then we discuss the deterministic model briefly, and establish the necessary conditions and the sufficient conditions for almost-sure extinction and persistence in mean for the stochastic system, where we show that the feedback controls can change the properties of the population systems significantly. Finally, numerical simulations are introduced to support the main results.

scholarly journals Impact of predator fear on two competing prey species

2021 ◽  
Vol 2 (1) ◽  
pp. 1-12
Author(s):  
Debasis Mukherjee
Keyword(s):  
Species Coexistence ◽  
Field Studies ◽  
Predation Rate ◽  
Uniform Persistence ◽  
Reproduction Rate ◽  
Predator Population ◽  
Predator Prey ◽  
Fear Effect ◽  
Prey Interaction ◽  
The Cost

Predator-prey interaction is a fundamental feature in the ecological system. The majority of studies have addressed how competition and predation affect species coexistence. Recent field studies on vertebrate has shown that fear of predators can influence the behavioural pattern of prey populations and reduce their reproduction. A natural question arises whether species coexistence is still possible or not when predator induce fear on competing species. Based on the above observation, we propose a mathematical model of two competing prey-one predator system with the cost of fear that affect not only the reproduction rate of both the prey population but also the predation rate of predator. To make the model more realistic, we incorporate intraspecific competition within the predator population. Biological justification of the model is shown through positivity and boundedness of solutions. Existence andstability of different boundary equilibria are discussed. Condition for the existence of coexistence equilibrium point is derived from showing uniform persistence. Local as well as a global stability criterion is developed. Bifurcation analysis is performed by choosing the fear effect as the bifurcation parameter of the model system. The nature of the limit cycle emerging through a Hopf bifurcation is indicated. Numerical experiments are carried out to test the theoretical results obtained from this model.

scholarly journals Stochastic dynamics of predator-prey interactions

PLoS ONE ◽  
2021 ◽  
Vol 16 (8) ◽  
pp. e0255880
Author(s):  
Abhyudai Singh
Keyword(s):  
Attack Rate ◽  
Stochastic Dynamics ◽  
Volterra Model ◽  
Reproduction Rate ◽  
Predator Population ◽  
Population Densities ◽  
Predator Prey ◽  
Density Dependent ◽  
Predator Prey Model ◽  
Predator Attack

The interaction between a consumer (such as, a predator or a parasitoid) and a resource (such as, a prey or a host) forms an integral motif in ecological food webs, and has been modeled since the early 20th century starting from the seminal work of Lotka and Volterra. While the Lotka-Volterra predator-prey model predicts a neutrally stable equilibrium with oscillating population densities, a density-dependent predator attack rate is known to stabilize the equilibrium. Here, we consider a stochastic formulation of the Lotka-Volterra model where the prey’s reproduction rate is a random process, and the predator’s attack rate depends on both the prey and predator population densities. Analysis shows that increasing the sensitivity of the attack rate to the prey density attenuates the magnitude of stochastic fluctuations in the population densities. In contrast, these fluctuations vary non-monotonically with the sensitivity of the attack rate to the predator density with an optimal level of sensitivity minimizing the magnitude of fluctuations. Interestingly, our systematic study of the predator-prey correlations reveals distinct signatures depending on the form of the density-dependent attack rate. In summary, stochastic dynamics of nonlinear Lotka-Volterra models can be harnessed to infer density-dependent mechanisms regulating predator-prey interactions. Moreover, these mechanisms can have contrasting consequences on population density fluctuations, with predator-dependent attack rates amplifying stochasticity, while prey-dependent attack rates countering to buffer fluctuations.

scholarly journals Ecosystem–atmosphere exchange of microorganisms in a Mediterranean grassland: new insights into microbial flux through a combined experimental-modeling approach

2017 ◽  
Author(s):  
Federico Carotenuto ◽  
Teodoro Georgiadis ◽  
Beniamino Gioli ◽  
Christel Leyronas ◽  
Cindy E. Morris ◽  
...  
Keyword(s):  
Deterministic Model ◽  
Point Of View ◽  
Cloud Formation ◽  
Training Dataset ◽  
Promising Tool ◽  
Biological Aerosols ◽  
Ice Nuclei ◽  
Lack Of Information ◽  
Bacteria And Fungi ◽  
The Impact

Abstract. Microbial aerosols (mainly composed by bacterial and fungal cells), may constitute up to 74 % of the total aerosol volume. These biological aerosols are relevant not only from the point of view of the dispersion of pathogenic species, but also due to the potential geochemical implications. Some bacteria and fungi may, in fact, serve as cloud condensation or ice nuclei, potentially affecting cloud formation and precipitation and are active at higher temperatures compared to their, much more intensively studied, inorganic counterparts. Simulations of the impact of microbial aerosols on climate are still hindered by the lack of information regarding their emissions from ground sources. This work tackles this knowledge gap by (i) applying a rigorous micrometeorological approach to the estimation of microbial net fluxes above a Mediterranean grassland and (ii) developing a deterministic model to estimate these emissions on the basis of a few easily recovered meteorological parameters (the PLAnET model). The grassland itself is characterized by an abundance of positive net microbial fluxes and the model proves to be a promising tool capable of capturing the day-to-day variability in microbial fluxes with a relatively small bias and sufficient accuracy. PLAnET is still in its infancy and will benefit from future campaigns extending the available training dataset as well as the inclusion of ever more complex and critical phenomena affecting the release of microbial aerosol (such as rainfall). The model itself is also adaptable as an emission module for dispersion and chemical transport models, allowing to further explore the impact of microbial aerosols on the atmosphere and climate.

scholarly journals The Dynamic Complexity of a Holling Type-IV Predator-Prey System with Stage Structure and Double Delays

2011 ◽  
Vol 2011 ◽  
pp. 1-19 ◽  
Author(s):  
Yakui Xue ◽  
Xiafeng Duan
Keyword(s):  
Time Delay ◽  
Equilibrium Point ◽  
Dynamical Behavior ◽  
Stage Structure ◽  
Point Of View ◽  
Maturation Time ◽  
Dynamic Complexity ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Type Iv

We invest a predator-prey model of Holling type-IV functional response with stage structure and double delays due to maturation time for both prey and predator. The dynamical behavior of the system is investigated from the point of view of stability switches aspects. We assume that the immature and mature individuals of each species are divided by a fixed age, and the mature predator only attacks the mature prey. Based on some comparison arguments, sharp threshold conditions which are both necessary and sufficient for the global stability of the equilibrium point of predator extinction are obtained. The most important outcome of this paper is that the variation of predator stage structure can affect the existence of the interior equilibrium point and drive the predator into extinction by changing the maturation (through-stage) time delay. Our linear stability work and numerical results show that if the resource is dynamic, as in nature, there is a window in maturation time delay parameters that generate sustainable oscillatory dynamics.

The dynamics of a delayed deterministic and stochastic epidemic SIR models with a saturated incidence rate

2018 ◽  
Vol 26 (4) ◽  
pp. 235-245 ◽  
Author(s):  
Modeste N’zi ◽  
Ilimidi Yattara
Keyword(s):  
Stochastic Model ◽  
White Noise ◽  
Incidence Rate ◽  
Deterministic Model ◽  
Contact Rate ◽  
Almost Sure Stability ◽  
Saturated Incidence Rate ◽  
Saturated Incidence ◽  
Global Positive Solution ◽  
Disease Free

AbstractWe treat a delayed SIR (susceptible, infected, recovered) epidemic model with a saturated incidence rate and its perturbation through the contact rate using a white noise. We start with a deterministic model and then add a perturbation on the contact rate using a white noise to obtain a stochastic model. We prove the existence and uniqueness of the global positive solution for both deterministic and stochastic delayed differential equations. Under suitable conditions on the parameters, we study the global asymptotic stability of the disease-free equilibrium of the deterministic model and the almost sure stability of the disease-free equilibrium of the stochastic model.

scholarly journals Rich Dynamics of a Predator-Prey System with Different Kinds of Functional Responses

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-19
Author(s):  
Kankan Sarkar ◽  
Subhas Khajanchi ◽  
Prakash Chandra Mali ◽  
Juan J. Nieto
Keyword(s):  
Growth Rate ◽  
Hopf Bifurcation ◽  
Dynamical Behavior ◽  
Prey Population ◽  
Uniform Persistence ◽  
Functional Responses ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Prey System

In this study, we investigate a mathematical model that describes the interactive dynamics of a predator-prey system with different kinds of response function. The positivity, boundedness, and uniform persistence of the system are established. We investigate the biologically feasible singular points and their stability analysis. We perform a comparative study by considering different kinds of functional responses, which suggest that the dynamical behavior of the system remains unaltered, but the position of the bifurcation points altered. Our model system undergoes Hopf bifurcation with respect to the growth rate of the prey population, which indicates that a periodic solution occurs around a fixed point. Also, we observed that our predator-prey system experiences transcritical bifurcation for the prey population growth rate. By using normal form theory and center manifold theorem, we investigate the direction and stability of Hopf bifurcation. The biological implications of the analytical and numerical findings are also discussed in this study.

Stochastic dynamics for the solutions of a modified Holling–Tanner model with random perturbation

2014 ◽  
Vol 25 (11) ◽  
pp. 1450105 ◽  
Author(s):  
Zhenjie Liu
Keyword(s):  
Stochastic System ◽  
Stochastic Dynamics ◽  
Deterministic Model ◽  
Random Perturbation ◽  
Sufficient Conditions ◽  
Unique Positive Solution ◽  
Initial Value ◽  
Environmental Forcing ◽  
Predator Prey ◽  
Predator Prey Model

In this paper, we consider a stochastic nonautonomous predator–prey model with modified Leslie–Gower and Holling II schemes in the presence of environmental forcing. The deterministic model is the modified Holling–Tanner model which is an extension of the classical Leslie–Gower model. We show that there is a unique positive solution to the stochastic system for any positive initial value. Sufficient conditions for strong persistence in mean and extinction to the stochastic system are established.

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