Modeling the Effect of Fear in a Prey–Predator System with Prey Refuge and Gestation Delay

2019 ◽  
Vol 29 (14) ◽  
pp. 1950195 ◽  
Author(s):  
Ankit Kumar ◽  
Balram Dubey
Keyword(s):  
Hopf Bifurcation ◽  
Prey Population ◽  
Reproduction Rate ◽  
Predator Population ◽  
Maximum Lyapunov Exponent ◽  
Stable Limit ◽  
Prey Refuge ◽  
Gestation Delay ◽  
Fear Effect ◽  
Predator System

Recently, some field experiments and studies show that predators affect prey not only by direct killing, they induce fear in prey which reduces the reproduction rate of prey species. Considering this fact, we propose a mathematical model to study the fear effect and prey refuge in prey–predator system with gestation time delay. It is assumed that prey population grows logistically in the absence of predators and the interaction between prey and predator is followed by Crowley–Martin type functional response. We obtained the equilibrium points and studied the local and global asymptotic behaviors of nondelayed system around them. It is observed from our analysis that the fear effect in the prey induces Hopf-bifurcation in the system. It is concluded that the refuge of prey population under a threshold level is lucrative for both the species. Further, we incorporate gestation delay of the predator population in the model. Local and global asymptotic stabilities for delayed model are carried out. The existence of stable limit cycle via Hopf-bifurcation with respect to delay parameter is established. Chaotic oscillations are also observed and confirmed by drawing the bifurcation diagram and evaluating maximum Lyapunov exponent for large values of delay parameter.

scholarly journals Impact of fear in a prey-predator system with herd behaviour

2021 ◽  
Vol 9 (1) ◽  
pp. 175-197
Author(s):  
Sangeeta Saha ◽  
Guruprasad Samanta
Keyword(s):  
Hopf Bifurcation ◽  
Death Rate ◽  
Birth Rate ◽  
Prey Species ◽  
Prey Population ◽  
Predator Population ◽  
Defense Strategy ◽  
Fear Effect ◽  
Herd Behaviour ◽  
Predator System

Abstract Fear of predation plays an important role in the growth of a prey species in a prey-predator system. In this work, a two-species model is formulated where the prey species move in a herd to protect themselves and so it acts as a defense strategy. The birth rate of the prey here is affected due to fear of being attacked by predators and so, is considered as a decreasing function. Moreover, there is another fear term in the death rate of the prey population to emphasize the fact that the prey may die out of fear of predator too. But, in this model, the function characterizing the fear effect in the death of prey is assumed in such a way that it is increased only up to a certain level. The results show that the system performs oscillating behavior when the fear coefficient implemented in the birth of prey is considered in a small amount but it changes its dynamics through Hopf bifurcation and becomes stable for a higher value of the coefficient. Regulating the fear terms ultimately makes an impact on the growth of the predator population as the predator is taken as a specialist predator here. The increasing value of the fear terms (either implemented in birth or death of prey) decrease the count of the predator population with time. Also, the fear implemented in the birth rate of prey makes a higher impact on the growth of the predator population than in the case of the fear-induced death rate.

A NONAUTONOMOUS MODEL FOR THE INTERACTIVE EFFECTS OF FEAR, REFUGE AND ADDITIONAL FOOD IN A PREY–PREDATOR SYSTEM

2021 ◽  
pp. 1-39
Author(s):  
NAZMUL SK ◽  
PANKAJ KUMAR TIWARI ◽  
YUN KANG ◽  
SAMARES PAL
Keyword(s):  
Growth Rate ◽  
Hopf Bifurcation ◽  
Seasonal Variations ◽  
Positive Periodic Solution ◽  
Oscillatory System ◽  
Interactive Effects ◽  
Predator Population ◽  
Prey Refuge ◽  
Additional Food ◽  
Predator System

The importance of fear, refuge and additional food is being increasingly recognized in recent studies, but their combined effects need to be explored. In this paper, we investigate the joint effects of these three ecologically important factors in a prey–predator system with Crowly–Martin type functional response. We find that the fear of predator significantly affects the densities of prey and predator populations whereas the presence of prey refuge and additional food for predator are recognized to have potential impacts to sustain prey and predator in the habitat, respectively. The fear of predator induces limit cycle oscillations while an oscillatory system becomes stable on increasing the refuge. The system first undergoes a supercritical Hopf-bifurcation and then a subcritical Hopf-bifurcation on increasing either the growth rate of prey or growth rate of predator due to additional food. Increasing the quality/quantity of additional food after a certain value causes extinction of prey species and rapid incline in the predator population. An extension is made in the model by considering the seasonal variations in the cost of fear of predator, prey refuge and growth rate of predator due to additional food. The nonautonomous model is shown to exhibit globally attractive positive periodic solution. Moreover, complex dynamics such as higher periodic solutions and bursting patterns are observed due to seasonal variations in the rate parameters.

scholarly journals Modelling of a two prey and one predator system with switching effect

2021 ◽  
Vol 9 (1) ◽  
pp. 90-113
Author(s):  
Sangeeta Saha ◽  
Guruprasad Samanta
Keyword(s):  
Steady State ◽  
Hopf Bifurcation ◽  
Stable State ◽  
Predation Rate ◽  
Prey Population ◽  
Predator Population ◽  
Prey Switching ◽  
Time Switching ◽  
Local Bifurcations ◽  
Predator System

Abstract Prey switching strategy is adopted by a predator when they are provided with more than one prey and predator prefers to consume one prey over others. Though switching may occur due to various reasons such as scarcity of preferable prey or risk in hunting the abundant prey. In this work, we have proposed a prey-predator system with a particular type of switching functional response where a predator feeds on two types of prey but it switches from one prey to another when a particular prey population becomes lower. The ratio of consumption becomes significantly higher in the presence of prey switching for an increasing ratio of prey population which satisfies Murdoch’s condition [15]. The analysis reveals that two prey species can coexist as a stable state in absence of predator but a single prey-predator situation cannot be a steady state. Moreover, all the population can coexist only under certain restrictions. We get bistability for a certain range of predation rate for first prey population. Moreover, varying the mortality rate of the predator, an oscillating system can be obtained through Hopf bifurcation. Also, the predation rate for the first prey can turn a steady-state into an oscillating system. Except for Hopf bifurcation, some other local bifurcations also have been studied here. The figures in the numerical simulation have depicted that, if there is a lesser number of one prey present in a system, then with time, switching to the other prey, in fact, increases the predator population significantly.

scholarly journals Dynamics of prey–predator model with strong and weak Allee effect in the prey with gestation delay

2020 ◽  
Vol 25 (3) ◽  
Author(s):  
Ankit Kumar ◽  
Balram Dubey
Keyword(s):  
Allee Effect ◽  
Threshold Value ◽  
Prey Population ◽  
Predator Population ◽  
Allee Effects ◽  
Normal Form Theory ◽  
Delayed Systems ◽  
Gestation Delay ◽  
The Status ◽  
Manifold Theory

This study proposes two prey–predator models with strong and weak Allee effects in prey population with Crowley–Martin functional response. Further, gestation delay of the predator population is introduced in both the models. We discussed the boundedness, local stability and Hopf-bifurcation of both nondelayed and delayed systems. The stability and direction of Hopfbifurcation is also analyzed by using Normal form theory and Center manifold theory. It is shown that species in the model with strong Allee effect become extinct beyond a threshold value of Allee parameter at low density of prey population, whereas species never become extinct in weak Allee effect if they are initially present. It is also shown that gestation delay is unable to avoiding the status of extinction. Lastly, numerical simulation is conducted to verify the theoretical findings. 

scholarly journals Dynamics of a Predator-Prey Population in the Presence of Resource Subsidy under the Influence of Nonlinear Prey Refuge and Fear Effect

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-38
Author(s):  
Sudeshna Mondal ◽  
G. P. Samanta ◽  
Juan J. Nieto
Keyword(s):  
Equilibrium Points ◽  
Time Lag ◽  
Sufficient Conditions ◽  
Food Sources ◽  
Prey Population ◽  
Predator Population ◽  
Input Rate ◽  
Prey Refuge ◽  
Predator Prey ◽  
The Impact

In this work, our aim is to investigate the impact of a non-Kolmogorov predator-prey-subsidy model incorporating nonlinear prey refuge and the effect of fear with Holling type II functional response. The model arises from the study of a biological system involving arctic foxes (predator), lemmings (prey), and seal carcasses (subsidy). The positivity and asymptotically uniform boundedness of the solutions of the system have been derived. Analytically, we have studied the criteria for the feasibility and stability of different equilibrium points. In addition, we have derived sufficient conditions for the existence of local bifurcations of codimension 1 (transcritical and Hopf bifurcation). It is also observed that there is some time lag between the time of perceiving predator signals through vocal cues and the reduction of prey’s birth rate. So, we have analyzed the dynamical behaviour of the delayed predator-prey-subsidy model. Numerical computations have been performed using MATLAB to validate all the analytical findings. Numerically, it has been observed that the predator, prey, and subsidy can always exist at a nonzero subsidy input rate. But, at a high subsidy input rate, the prey population cannot persist and the predator population has a huge growth due to the availability of food sources.

THE ROLE OF ADDITIONAL FOOD IN A PREDATOR–PREY MODEL WITH A PREY REFUGE

2016 ◽  
Vol 24 (02n03) ◽  
pp. 345-365 ◽  
Author(s):  
SUDIP SAMANTA ◽  
RIKHIYA DHAR ◽  
IBRAHIM M. ELMOJTABA ◽  
JOYDEV CHATTOPADHYAY
Keyword(s):  
Stable Equilibrium ◽  
Predator Population ◽  
Stable Limit ◽  
Prey Refuge ◽  
Additional Food ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
The Stability ◽  
The Impact

In this paper, we propose and analyze a predator–prey model with a prey refuge and additional food for predators. We study the impact of a prey refuge on the stability dynamics, when a constant proportion or a constant number of prey moves to the refuge area. The system dynamics are studied using both analytical and numerical techniques. We observe that the prey refuge can replace the predator–prey oscillations by a stable equilibrium if the refuge size crosses a threshold value. It is also observed that, if the refuge size is very high, then the extinction of the predator population is certain. Further, we observe that enhancement of additional food for predators prevents the extinction of the predator and also replaces the stable limit cycle with a stable equilibrium. Our results suggest that additional food for the predators enhances the stability and persistence of the system. Extensive numerical experiments are performed to illustrate our analytical findings.

Age-selective harvesting in a delayed predator–prey model with fear effect

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Ashok Mondal ◽  
Amit K. Pal
Keyword(s):  
Hopf Bifurcation ◽  
Predator Population ◽  
Ecological Management ◽  
Predator Prey ◽  
Selective Harvesting ◽  
Predator Prey Model ◽  
Elapsed Time ◽  
Fear Effect ◽  
Prey Model ◽  
Fear Effects

Abstract In this article, we discussed the dynamic behavior of a delay-induced harvested predator–prey model with fear effects (perceived by the prey). We then considered selective harvesting terms for both species which provide some fixed elapsed time to the prey and for the predator population before they are harvested. In other words, we are limiting the harvesting of species below a certain age so that they can grow to a certain specific size or age and thus protect juvenile populations. Reproduction of the prey population can also be greatly impeded due to the influence of the fear effect. The consideration of selective harvesting together with the effect of fear on the proposed system to show stable coexistence to the oscillatory mode and vice versa via Hopf-bifurcation. For better ecological management of the community, our study reveals the fact that collection delays and intensities should be maintained. Numerical simulations were performed to validate our analytical results.

GLOBAL BEHAVIOR AND HOPF BIFURCATION OF A STAGE-STRUCTURED PREY–PREDATOR MODEL WITH MATURATION DELAY FOR PREY AND GESTATION DELAY FOR PREDATOR

2015 ◽  
Vol 23 (01) ◽  
pp. 57-77 ◽  
Author(s):  
KUNWER SINGH JATAV ◽  
JOYDIP DHAR
Keyword(s):  
Hopf Bifurcation ◽  
Equilibrium Points ◽  
Sufficient Conditions ◽  
Biological Properties ◽  
Life Cycles ◽  
Comparison Theorems ◽  
Prey Population ◽  
Gestation Delay ◽  
Boundary Equilibrium ◽  
Age Structured

Age structured models are important for the dynamics and evolution of many insect populations. For such models the rates of survival, growth and reproduction depend on the age or developmental stage of individuals in the population. For example, the life cycles of many species are composed of at least two stages, immature individuals (immatures) and mature individuals (matures) which may exhibit fundamentally different developmental, morphological and behavioral characteristics. In this paper, we incorporate these biological properties into a predator–prey population model. Our model is stage-structured for the prey and contains maturation and gestation delays. Using comparison theorems, we derive sufficient conditions for dynamical system permanence. We found equilibrium points exhibit switching with boundary equilibrium points through a mechanism of decreasing maturation of the prey population. Furthermore, an analysis of the corresponding characteristic equations was performed to determine local stability of equilibria. Choosing the gestation delay as a bifurcation parameter, we were able to compute a critical value for the existence of small amplitude oscillation of population densities (i.e., a Hopf bifurcation). Sufficient conditions for the global stability of all non-negative equilibria were obtained using Kamke comparison theorems and a newly developed iteration technique. Numerical simulations were performed to illustrate the theoretical results.

Influence of the Fear Effect on a Holling Type II Prey–Predator System with a Michaelis–Menten Type Harvesting

2021 ◽  
Vol 31 (14) ◽  
Author(s):  
Binfeng Xie ◽  
Zhengce Zhang ◽  
Na Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Oscillation ◽  
Positive Equilibrium ◽  
Type Ii ◽  
Bifurcation Parameter ◽  
Fear Effect ◽  
Existence And Stability ◽  
Positive Equilibrium Point ◽  
The Stability ◽  
Predator System

In this work, a prey–predator system with Holling type II response function including a Michaelis–Menten type capture and fear effect is put forward to be studied. Firstly, the existence and stability of equilibria of the system are discussed. Then, by considering the harvesting coefficient as bifurcation parameter, the occurrence of Hopf bifurcation at the positive equilibrium point and the existence of limit cycle emerging through Hopf bifurcation are proved. Furthermore, through the analysis of fear effect and capture item, we find that: (i) the fear effect can either stabilize the system by excluding periodic solutions or destroy the stability of the system and produce periodic oscillation behavior; (ii) increasing the level of fear can reduce the final number of predators, but not lead to extinction; (iii) the harvesting coefficient also has significant influence on the persistence of the predator. Finally, numerical simulations are presented to illustrate the results.

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