Stability and Hopf-bifurcation in a general Gauss type two-prey and one-predator system

Deka, Bhupen; Patra, Atasi; Tushar, Jai; Dubey, B.

doi:10.1016/j.apm.2016.01.018

Cited by 22 publications

(9 citation statements)

References 21 publications

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“…Deka et al [14], Fujii [12] and Takeuchi and Adachi [13] addressed an ecological system with the same type of species, but no interfering time to competitive response and anti-predator behavior for obtaining coexistence results. Finally, we note that if competing takes time to both competing species, then competition pressure becomes low, which enhances the coexistence when there is no predator.…”

Section: Discussionmentioning

confidence: 99%

“…They remarked that chaotic motion arises from periodic motion when one of two prey has greater competitive abilities than the other and predator mediated coexistence is possible depending on the preferences of a predator and competitive abilities of two prey. Deka et al [14] studied the effect of predation on two competing prey species in the general Gauss type model.…”

Section: Introductionmentioning

confidence: 99%

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Stability and bifurcation of a two competing prey-one predator system with anti-predator behavior

Mukherjee

2022

Jambura J. Biomath.

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This article considers the impact of competitive response to interfering time and anti-predator behavior of a three species system in which one predator consumes both the competing prey species. Here one of the competing species shows anti-predator behavior. We have shown that its solutions are non-negative and bounded. Further, we analyze the existence and stability of all the feasible equilibria. Conditions for uniform persistence of the system are derived. Applying Bendixson’s criterion for high-dimensional ordinary differential equations, we prove that the coexistence equilibrium point is globally stable under specific conditions. The system admits Hopf bifurcation when anti-predator behavior rate crosses a critical value. Analytical results are verified numerically.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Stability and bifurcation of a two competing prey-one predator system with anti-predator behavior

Mukherjee

2022

Jambura J. Biomath.

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“…We are able to carry out stability analysis of the positive equilibrium, similar to that of Deka et al (2016). The Jacobian is given by (A.4).…”

Section: Appendix A2 Secondary Prey-free Equilibriummentioning

confidence: 99%

Hybrid approach to modeling spatial dynamics of systems with generalist predators

Fussell

Krause

Gorder

2019

Journal of Theoretical Biology

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We consider hybrid spatial modeling approaches for ecological systems with a generalist predator utilizing a prey and either a second prey or an allochthonous resource. While spatial dispersion of populations is often modeled via stepping-stone (discrete spatial patches) or continuum (one connected spatial domain) formulations, we shall be interested in hybrid approaches which we use to reduce the dimension of certain components of the spatial domain, obtaining either a continuum model of varying spatial dimensions, or a mixed stepping-stone-continuum model. This approach results in models consisting of partial differential equations for some of the species which are coupled via reactive boundary conditions to lower dimensional partial differential equations or ordinary differential equations for the other species. In order to demonstrate the use of this approach, we consider two case studies. In the first case study, we consider a one-predator two-prey interaction between beavers, wolves and white-tailed deer in Voyageurs National Park. In the second case study, we consider predator-prey-allochthonous resource interactions between bears, berries and salmon on Kodiak Island. For each case study, we compare the results from the hybrid modeling approach with corresponding stepping-stone and continuum model results, highlighting benefits and limitations of the method. In some cases, we find that the hybrid modeling approach allows for solutions which are easier to simulate (akin to stepping-stone models) while maintaining seemingly more realistic spatial dynamics (akin to full continuum models).

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“…It is well known that gestation delay, as a time lag, plays a complicated role on the 2 Complexity dynamics of the prey-predator system. Thus, the considered system can be governed by the following delay differential equations [2,5]:…”

Section: The Mathematical Modelmentioning

confidence: 99%

“…Kar and Batabyal [4] have studied a two-prey one-predator system in the presence of time delay and derived the conditions for persistence and global stability of the model system. Recently, Deka et al [5] considered a general Gause-type two-prey one-predator model system and studied the effect of predation on two competing prey species. The authors observed Hopf bifurcation by taking the death rate of the predator population as a bifurcation parameter.…”

Section: Introductionmentioning

confidence: 99%

The Gestation Delay: A Factor Causing Complex Dynamics in Gause-Type Competition Models

et al. 2018

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In this paper, we consider a Gause-type model system consisting of two prey and one predator. Gestation period is considered as the time delay for the conversion of both the prey and predator. Bobcats and their primary prey rabbits and squirrels, found in North America and southern Canada, are taken as an example of an ecological system. It has been observed that there are stability switches and the system becomes unstable due to the effect of time delay. Positive invariance, boundedness, and local stability analysis are studied for the model system. Conditions under which both delayed and nondelayed model systems remain globally stable are found. Criteria which guarantee the persistence of the delayed model system are derived. Conditions for the existence of Hopf bifurcation at the nonzero equilibrium point of the delayed model system are also obtained. Formulae for the direction, stability, and period of the bifurcating solution are conducted using the normal form theory and center manifold theorem. Numerical simulations have been shown to analyze the effect of each of the parameters considered in the formation of the model system on the dynamic behavior of the system. The findings are interesting from the application point of view.

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Stability and Hopf-bifurcation in a general Gauss type two-prey and one-predator system

Cited by 22 publications

References 21 publications

Stability and bifurcation of a two competing prey-one predator system with anti-predator behavior

Stability and bifurcation of a two competing prey-one predator system with anti-predator behavior

Hybrid approach to modeling spatial dynamics of systems with generalist predators

The Gestation Delay: A Factor Causing Complex Dynamics in Gause-Type Competition Models

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