Stationary patterns induced by self- and cross-diffusion in a Beddington–DeAngelis predator–prey model

Guin, Lakshmi Narayan; Mondal, Benukar; Chakravarty, S.

doi:10.1007/s40435-016-0281-7

Cited by 18 publications

(12 citation statements)

References 26 publications

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“…One of the novel features about our model is to account for the universally prevalent intra-specific competition in the consumer growth dynamics [21]. This intraspecific competition is assumed to induce an additional increased death rate, which is proportional to the square of the adult population [12,13,15]. We assume that prey and adult predator are harvested as CPUE hypothesis, and juvenile predators are not harvested.…”

Section: Mathematical Model and Its Qualitative Propertiesmentioning

confidence: 99%

“…The effect of intra-specific interference among predators has been studied in a preypredator model with Holling-type II functional response in [13], with Holling-type III functional response in [12] and with Beddington-DeAngelis-type functional response in [15]. In these three studies, spatiotemporal dynamics of the system are also investigated.…”

Section: Introductionmentioning

confidence: 99%

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Optimal harvesting policy of a prey–predator model with Crowley–Martin-type functional response and stage structure in the predator

Dubey

Agarwal

Kumar

2018

NAMC

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In this paper, a three-dimensional dynamical model consisting of a prey, a mature predator, and an immature predator is proposed and analysed. The interaction between prey and mature predator is assumed to be of the Crowley–Martin type, and both the prey and mature predator are harvested according to catch-per-unit-effort (CPUE) hypothesis. Steady state of the system is obtained, stability analysis (local and global both) are discussed to explore the long-time behaviour of the system. The optimal harvesting policy is also discussed with the help of Pontryagin's maximum principle. The harvesting effort is taken as an effective control instrument to preserve prey and predator and to maintain them at an optimal level.

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Section: Mathematical Model and Its Qualitative Propertiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Optimal harvesting policy of a prey–predator model with Crowley–Martin-type functional response and stage structure in the predator

Dubey

Agarwal

Kumar

2018

NAMC

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“…Baghel and Dhar ( 2014 ) performed a high-order stability analysis in a three-species food chain model. Guin et al ( 2017 ) derived essential condition for Hopf and Turing bifurcations in a predator–prey model. Also, the study complied with self and cross-diffusion patterns.…”

Section: Introductionmentioning

confidence: 99%

Appearance of Temporal and Spatial Chaos in an Ecological System: A Mathematical Modeling Study

Raw

Mishra

Sarangi

et al. 2021

Iran J Sci Technol Trans Sci

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The ecological theory of species interactions rests largely on the competition, interference, and predator–prey models. In this paper, we propose and investigate a three-species predator–prey model to inspect the mutual interference between predators. We analyze boundedness and Kolmogorov conditions for the non-spatial model. The dynamical behavior of the system is analyzed by stability and Hopf bifurcation analysis. The Turing instability criteria for the Spatio-temporal system is estimated. In the numerical simulation, phase portrait with time evolution diagrams shows periodic and chaotic oscillations. Bifurcation diagrams show the very rich and complex dynamical behavior of the non-spatial model. We calculate the Lyapunov exponent to justify the dynamics of the non-spatial model. A variety of patterns like interference, spot, and stripe are observed with special emphasis on Beddington–DeAngelis function response. These complex patterns explore the beauty of the spatio-temporal model and it can be easily related to real-world biological systems.

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“…Mathematical ecology aims at the mathematical depiction, behaviour, and modelling of complex ecosystems using a choice of applied mathematical techniques and tools to get a better understanding of the system [3,9,[11][12][13]. Predator-prey relationship plays a significant role in regulating the numbers of prey and predators in the environment of complex ecosystem [14,15]. e harvesting used by both the species has a strong impact on the system dynamics of the predator-prey interaction.…”

Section: Introductionmentioning

confidence: 99%

Dynamical Analysis of a Beddington–DeAngelis Interacting Species System with Prey Harvesting

Guin

Murmu

Baek

et al. 2020

Mathematical Problems in Engineering

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A reaction–diffusion interacting species system with Beddington–DeAngelis functional response that has been proposed in the environment of mathematical ecology, which provides the rise to spatial pattern formation, is investigated and associated with the models of deterministic dynamics. The dynamical behaviour of a generalist predator–prey system with linear harvesting of each species and predator-dependent functional response is fully analyzed. Conditions of stability behaviour of the interior equilibrium point are established properly. Furthermore, we have recognized that the unique positive equilibrium point of the system is globally stable via appropriate Lyapunov function structure, which signifies that appropriate harvesting has no impact on the persistence property of the harvesting system. Also, we establish the conditions for the existence of bifurcation phenomena including a saddle-node bifurcation and a Hopf bifurcation. Subsequently, complete analysis regarding the impact of harvesting is carried out, and an interesting decision is that under some appropriate constraints, harvesting has immense impact on the final size of the interacting species. In addition, in accordance with Turing’s ideas on morphogenesis , our analysis shows that harvesting effort in a reaction–diffusion predator–prey system plays a vital function for geological conservation of interacting species. Finally, we discuss sufficient conditions for the existence of bionomic equilibrium point and the optimal harvesting policy attained by using the Pontryagin maximal principle.

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Stationary patterns induced by self- and cross-diffusion in a Beddington–DeAngelis predator–prey model

Cited by 18 publications

References 26 publications

Optimal harvesting policy of a prey–predator model with Crowley–Martin-type functional response and stage structure in the predator

Optimal harvesting policy of a prey–predator model with Crowley–Martin-type functional response and stage structure in the predator

Appearance of Temporal and Spatial Chaos in an Ecological System: A Mathematical Modeling Study

Dynamical Analysis of a Beddington–DeAngelis Interacting Species System with Prey Harvesting

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