Predator-prey relationships: Indiscriminate predation

Saleem, Muhammad

doi:10.1007/bf00275220

Cited by 8 publications

(6 citation statements)

References 11 publications

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“…As we known, when age structure is introduced into predator-prey interactions, population models become remarkably complex. Hence, usually A C C E P T E D M A N U S C R I P T people ignore age structure in one species, either in predator or in prey (Gurtin and Levine [11], Levine [17], Saleem [29], Saleem [3]). …”

Section: Accepted M M a N U mentioning

confidence: 97%

“…Tranditionally most predator-prey studies have focused on interacting species without age structure. However, as the importance of age structure in populations has become more widely recognized, in recent years there has been a rapidly growing literature dealing with various aspects of interacting populations with age structure (Gurtin and Levine [11], Cushing [5], [6], Cushing and Saleem [3], Pruss [26], Levine [17], Saleem [29], [30], Webb [34], Li [18]). As we known, when age structure is introduced into predator-prey interactions, population models become remarkably complex.…”

Section: Accepted M M a N U mentioning

confidence: 98%

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Hopf bifurcation for a predator–prey model with age structure

Tang

Liu

2016

Applied Mathematical Modelling

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Highlights• Consider a predator-prey model with age structure • Rewrite the age structured model as a non-densely defined Cauchy problem.• The stability and Hopf bifurcation are studied.• The numerical simulations support our results. AbstractA predator-prey model is investigated in which the predator population is assumed to have an age structure. We formulate the model as an abstract non-densely defined Cauchy problem and derive the condition which guarantees the existence and uniqueness of positive age-dependent equilibrium for the model. Bifurcation analysis indicates that the predator-prey system with age structure exhibits Hopf bifurcation which is the main result of this paper. Numerical simulations are given to illustrate the given results.

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Section: Accepted M M a N U mentioning

confidence: 97%

Section: Accepted M M a N U mentioning

confidence: 98%

Hopf bifurcation for a predator–prey model with age structure

Tang

Liu

2016

Applied Mathematical Modelling

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“…Moreover, such a model structure is similar to the classical predator-prey interaction systems, but is different. The nonlinear dynamics of age-structured predator-prey population models have been studied by many researchers, see for example, Cushing [2,3], Cushing and Saleem [4], Gurtin and Levine [6], Levine [9], Li [10], Saleem [23], and Venturino [25], and various interesting asymptotical behaviors including bifurcation have been observed. It will be very interesting to apply the general Hopf bifurcation theorem in Liu et al [11] to study Hopf bifurcations in predator-prey population models when both predator and prey species are age-structured.…”

Section: Assume In Addition Thatmentioning

confidence: 99%

Oscillations in age-structured models of consumer-resource mutualisms

Liu¹,

Magal²,

Ruan³

2015

DCDS-B

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In consumer-resource interactions, a resource is regarded as a biotic population that helps to maintain the population growth of its consumer, whereas a consumer exploits a resource and then reduces its growth rate. Bi-directional consumer-resource interactions describe the cases where each species acts as both a consumer and a resource of the other, which is the basis of many mutualisms. In uni-directional consumer-resource interactions one species acts as a consumer and the other as a material and/or energy resource while neither acts as both. In this paper we consider an age-structured model for uni-directional consumer-resource mutualisms in which the consumer species has both positive and negative effects on the resource species, while the resource has only a positive effect on the consumer. Examples include a predator-prey system in which the prey is able to kill or consume predator eggs or larvae and the insect pollinator and the host plant relationship in which the plants provide food, seeds, nectar and other resources for the pollinators while the pollinators have both positive and negative effects on the plants. By carrying out local analysis and bifurcation analysis of the model, we discuss the stability of the positive equilibrium and show that under some conditions a non-trivial periodic solution through Hopf bifurcation appears when the maturation parameter passes through some critical values.2010 Mathematics Subject Classification. Primary: 35B32, 35Q92; Secondary: 92D30.

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“…For equilibrium analysis of an age-structured competition model, see [20,36]. The existence results and formulation of age-structured models were introduced by Gurtin and MacCamy [21] and Venturino [43].…”

Section: Introductionmentioning

confidence: 99%

“…Barbu and Iannelli [6] studied the mathematical theory behind age-structured populations. Brauer [9], Li [26], and Saleem [36] provided insight into nonlinear age-dependent populations and predator prey dynamics. Fister and Lenhart [17] investigated the existence of solutions to a predator prey age-structured model with control.…”

Section: Introductionmentioning

confidence: 99%

An age-structured model for the spread of epidemic cholera: Analysis and simulation

Alexanderian¹,

Gobbert²,

Fister³

et al. 2011

Nonlinear Analysis: Real World Applications

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Occasional outbreaks of cholera epidemics across the world demonstrate that the disease continues to pose a public health threat. Traditional models for the spread of infectious diseases are based on systems of ordinary differential equations. Since disease dynamics such as vaccine efficacy and the risk for contracting cholera depend on the age of the humans, an age-structured model offers additional insights and the possibility to study the effects of treatment options. The investigated model is given as a system of hyperbolic (first order) partial differential equations in combination with ordinary differential equations. First, using a representation from the method of characteristics and a fixed point argument, we prove the existence and uniqueness of a solution to our nonlinear system. Then we present a finite difference approximation to the model and study the effect of high and low rates of shedding of cholera vibrios on the dynamics of the spread of the disease. The simulations demonstrate the explosive nature of cholera outbreaks that is observed in reality. The contrast of results for high and low rates of shedding of vibrios suggest a possible underlying cause for this effect.

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Predator-prey relationships: Indiscriminate predation

Cited by 8 publications

References 11 publications

Hopf bifurcation for a predator–prey model with age structure

Hopf bifurcation for a predator–prey model with age structure

Oscillations in age-structured models of consumer-resource mutualisms

An age-structured model for the spread of epidemic cholera: Analysis and simulation

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