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.

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.

A linear birth-and-death predator–prey process

1995 ◽  
Vol 32 (01) ◽  
pp. 274-277
Author(s):  
John Coffey
Keyword(s):  
Death Rate ◽  
Birth Rate ◽  
Prey Population ◽  
Death Process ◽  
Predator Population ◽  
Birth And Death Process ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model

A new stochastic predator-prey model is introduced. The predator population X(t) is described by a linear birth-and-death process with birth rate λ 1 X and death rate μ 1 X. The prey population Y(t) is described by a linear birth-and-death process in which the birth rate is λ 2 Y and the death rate is . It is proven that and iff

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.

A linear birth-and-death predator–prey process

1995 ◽  
Vol 32 (1) ◽  
pp. 274-277 ◽  
Author(s):  
John Coffey
Keyword(s):  
Death Rate ◽  
Birth Rate ◽  
Prey Population ◽  
Death Process ◽  
Predator Population ◽  
Birth And Death Process ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model

A new stochastic predator-prey model is introduced. The predator population X(t) is described by a linear birth-and-death process with birth rate λ1X and death rate μ1X. The prey population Y(t) is described by a linear birth-and-death process in which the birth rate is λ2Y and the death rate is . It is proven that and iff

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.

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.

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.

scholarly journals A prey-predator system with herd behaviour of prey in a rapidly fluctuating environment

2019 ◽  
Vol 1 (1) ◽  
pp. 16-26
Author(s):  
G.P. SAMANTA ◽  
A. Mondal ◽  
D. Sahoo ◽  
P. Dolai
Keyword(s):  
Statistical Theory ◽  
Differential Equations ◽  
Random Environment ◽  
Prey Density ◽  
Group Living ◽  
Prey Population ◽  
Square Root ◽  
Herd Behaviour ◽  
Predator System ◽  
Perturbation Approximation

A statistical theory of non-equilibrium fluctuation in damped Volterra-Lotka prey-predator system where prey population lives in herd in a rapidly fluctuating random environment has been presented. The method is based on the technique of perturbation approximation of non-linear coupled stochastic differential equations. The characteristic of group-living of prey population has been emphasized using square root of prey density in the functional response.

Birth Control: Reversal or Postponement?

1960 ◽  
Vol 3 ◽  
pp. 59-73 ◽  
Author(s):  
Leo A. Orleans
Keyword(s):  
Natural Resources ◽  
Birth Control ◽  
Death Rate ◽  
Birth Rate ◽  
Natural Increase ◽  
Vital Rates ◽  
The People ◽  
The World ◽  
Communist China ◽  
Annual Population

Whereas throughout most of the world the results of the 1953 censusregistration of Communist China, reporting a population of 582·6 million, evoked anxiety and even alarm, the Communists expressed only pride and overwhelming confidence. As a people “liberated from the oppressive chains of capitalism,” Communist leaders felt that their horizons were unlimited and that feeding and caring for a population of this size presented no problems under a system in which people are “the most precious of all categories of capital.” The simultaneous release of vital rates which indicated a birth rate of 37 per thousand population and a death rate of 17 per thousand, further stressed the “great vitality of the people of new China.” The 2 per cent, natural increase (excess of births over deaths), resulting in an annual population growth of some 12 million, was declared, in line with Marxist doctrine, to be an asset in a country with vast new lands and unexploited natural resources, where additional people create additional wealth.

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