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.

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.

scholarly journals Dynamics of Predator-Prey Community with Age Structures and Its Changing Due To Harvesting

2020 ◽  
Vol 15 (1) ◽  
pp. 73-92
Author(s):  
G.P. Neverova ◽  
O.L. Zhdanova ◽  
E.Ya. Frisman
Keyword(s):  
Community Dynamics ◽  
Initial Conditions ◽  
Reproductive Potential ◽  
Stable State ◽  
Stability Domain ◽  
Prey Population ◽  
Dynamic Mode ◽  
Predator Population ◽  
Time Model ◽  
Predator Prey

The paper studies dynamic modes of discrete-time model of structured predator-prey community like “arctic fox – rodent” and changing its dynamic modes due to interspecific interaction. We paid special attention to the analysis of situations in which changes in the dynamic modes are possible. In particularly, 3-cycle emerging in prey population can result in predator extinction. Moreover, this solution corresponding to an incomplete community simultaneously coexists with the solution describing dynamics of complete community, which can be both stable and unstable. The anthropogenic impact on the community dynamics is studied, that is realized as harvest of some part of predator or prey population. It is shown that prey harvesting leads to expansion of parameter space domain with non-trivial stable numbers of community populations. In this case, the prey harvest has little effect on the predator dynamics; changes are mainly associated with multistability areas. In particular, the multistability domain narrows, in which changing initial conditions leads to different dynamic regimes, such as the transition to a stable state or periodic oscillations. As a result, community dynamics becomes more predictable. It is shown that the dynamics of prey population is sensitive to its harvesting. Even a small harvest rate results in disappearance of population size fluctuations: the stable state captures the entire phase space in multistability areas. In the case of the predator population harvest, stability domain of the nontrivial fixed point expands along the parameter of the predator birth rate. Accordingly, a case where predator determines the prey population dynamics is possible only at high values of predator reproductive potential. It is shown that in the case of predator harvest, a change in the community dynamic mode is possible because of a shifting dynamic regime in the prey population initiating the same nature fluctuations in the predator population. The dynamic regimes emerging in the community models with and without harvesting were compared.

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.

BIFURCATION ANALYSIS FOR A ONE PREDATOR AND TWO PREY MODEL WITH PREY-TAXIS

2021 ◽  
Vol 29 (02) ◽  
pp. 495-524 ◽  
Author(s):  
EVAN C. HASKELL ◽  
JONATHAN BELL
Keyword(s):  
Steady State ◽  
Hopf Bifurcation ◽  
Pattern Formation ◽  
Temporal Pattern ◽  
Predator Population ◽  
Asymptotically Stable ◽  
Interaction Dynamics ◽  
Spatio Temporal ◽  
Population Interaction ◽  
Time Periodic

This paper concerns spatio-temporal pattern formation in a model for two competing prey populations with a common predator population whose movement is biased by direct prey-taxis mechanisms. By pattern formation, we mean the existence of stable, positive non-constant equilibrium states, or nontrivial stable time-periodic states. The taxis can be either repulsive or attractive and the population interaction dynamics is fairly general. Both types of pattern formation arise as one-parameter bifurcating solution branches from an unstable constant stationary state. In the absence of our taxis mechanism, the coexistence positive steady state, under suitable conditions, is locally asymptotically stable. In the presence of a sufficiently strong repulsive prey defense, pattern formation will develop. However, in the attractive taxis case, the attraction needs to be sufficiently weak for pattern formation to develop. Our method is an application of the Crandall–Rabinowitz and the Hopf bifurcation theories. We establish the existence of both types of branches and develop expressions for determining their stability.

scholarly journals Dynamics of Predator-Prey Community with Age Structures and Its Changing Due To Harvesting

2020 ◽  
Vol 15 (1) ◽  
pp. 73-92
Author(s):  
G.P. Neverova ◽  
O.L. Zhdanova ◽  
E.Ya. Frisman
Keyword(s):  
Community Dynamics ◽  
Initial Conditions ◽  
Reproductive Potential ◽  
Stable State ◽  
Stability Domain ◽  
Prey Population ◽  
Dynamic Mode ◽  
Predator Population ◽  
Time Model ◽  
Predator Prey

The paper studies dynamic modes of discrete-time model of structured predator-prey community like “arctic fox – rodent” and changing its dynamic modes due to interspecific interaction. We paid special attention to the analysis of situations in which changes in the dynamic modes are possible. In particularly, 3-cycle emerging in prey population can result in predator extinction. Moreover, this solution corresponding to an incomplete community simultaneously coexists with the solution describing dynamics of complete community, which can be both stable and unstable. The anthropogenic impact on the community dynamics is studied, that is realized as harvest of some part of predator or prey population. It is shown that prey harvesting leads to expansion of parameter space domain with non-trivial stable numbers of community populations. In this case, the prey harvest has little effect on the predator dynamics; changes are mainly associated with multistability areas. In particular, the multistability domain narrows, in which changing initial conditions leads to different dynamic regimes, such as the transition to a stable state or periodic oscillations. As a result, community dynamics becomes more predictable. It is shown that the dynamics of prey population is sensitive to its harvesting. Even a small harvest rate results in disappearance of population size fluctuations: the stable state captures the entire phase space in multistability areas. In the case of the predator population harvest, stability domain of the nontrivial fixed point expands along the parameter of the predator birth rate. Accordingly, a case where predator determines the prey population dynamics is possible only at high values of predator reproductive potential. It is shown that in the case of predator harvest, a change in the community dynamic mode is possible because of a shifting dynamic regime in the prey population initiating the same nature fluctuations in the predator population. The dynamic regimes emerging in the community models with and without harvesting were compared.

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.

Analytical Study of the Existence of a Hopf Bifurcation in the Tumor Cell Growth Model with Time Delay

2021 ◽  
Vol 3 (1) ◽  
pp. 20-28
Author(s):  
A. Yusnaeni ◽  
Kasbawati Kasbawati ◽  
Toaha Syamsuddin
Keyword(s):  
Differential Equation ◽  
Steady State ◽  
Immune System ◽  
Hopf Bifurcation ◽  
Delay Differential Equation ◽  
Immune Cells ◽  
Steady State Solution ◽  
State Solution ◽  
Activation Process ◽  
Delay Differential

AbstractIn this paper, we study a mathematical model of an immune response system consisting of a number of immune cells that work together to protect the human body from invading tumor cells. The delay differential equation is used to model the immune system caused by a natural delay in the activation process of immune cells. Analytical studies are focused on finding conditions in which the system undergoes changes in stability near a tumor-free steady-state solution. We found that the existence of a tumor-free steady-state solution was warranted when the number of activated effector cells was sufficiently high. By considering the lag of stimulation of helper cell production as the bifurcation parameter, a critical lag is obtained that determines the threshold of the stability change of the tumor-free steady state. It is also leading the system undergoes a Hopf bifurcation to periodic solutions at the tumor-free steady-state solution.Keywords: tumor–immune system; delay differential equation; transcendental function; Hopf bifurcation. AbstrakDalam makalah ini, dikaji model matematika dari sistem respon imun yang terdiri dari sejumlah sel imun yang bekerja sama untuk melindungi tubuh manusia dari invasi sel tumor. Persamaan diferensial tunda digunakan untuk memodelkan sistem kekebalan yang disebabkan oleh keterlambatan alami dalam proses aktivasi sel-sel imun. Studi analitik difokuskan untuk menemukan kondisi di mana sistem mengalami perubahan stabilitas di sekitar solusi kesetimbangan bebas tumor. Diperoleh bahwa solusi kesetimbangan bebas tumor dijamin ada ketika jumlah sel efektor yang diaktifkan cukup tinggi. Dengan mempertimbangkan tundaan stimulasi produksi sel helper sebagai parameter bifurkasi, didapatkan lag kritis yang menentukan ambang batas perubahan stabilitas dari solusi kesetimbangan bebas tumor. Parameter tersebut juga mengakibatkan sistem mengalami percabangan Hopf untuk solusi periodik pada solusi kesetimbangan bebas tumor.Kata kunci: sistem tumor–imun; persamaan differensial tundaan; fungsi transedental; bifurkasi Hopf.

A PREY-PREDATOR MODEL WITH DIFFUSION AND A SUPPLEMENTARY RESOURCE FOR THE PREY IN A TWO‐PATCH ENVIRONMENT

2005 ◽  
Vol 9 (1) ◽  
pp. 9-24 ◽  
Author(s):  
J. Dhar
Keyword(s):  
Steady State ◽  
Asymptotic Stability ◽  
Prey Species ◽  
Steady State Solution ◽  
State Solution ◽  
Prey Population ◽  
Predator Species ◽  
Additional Resource ◽  
Linear And Nonlinear ◽  
Predator Model

In this paper, a prey‐predator dynamics, where the predator species partially depends upon the prey species, in a two patch habitat with diffusion and there is a non‐diffusing additional resource for the prey population, is modeled and analyzed. It is shown, that there exists a positive, monotonic, continuous steady state solution with continuous matching at the interface for both the species separately. Further, we obtain conditions for asymptotic stability for both linear and nonlinear cases. Šiame straipsnyje modeliuojama ir analizuojama plešr‐unu ir auku dinamika, laikant, kad plešr-unu populiacija dalinai priklauso nuo auku skačiaus. Areala sudaro dvi sritys, kuriose vyksta populiaciju individu difuzija, be to, aukoms yra išskirtas nedifunduojantis resursas. Irodyta, kad egzistuoja teigiamas, monotoniškas, tolydus stacionarusis sprendinys, tenkinantis tolydumo salyga abiems populiacijoms atskirai. Gautos asimptotinio stabilumo salygos tiesiniu ir netiesiniu atvejais.

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