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Global dynamics of a diffusive SIR epidemic model with saturated incidence rate and discontinuous treatments. INTERNATIONAL JOURNAL OF DYNAMICS AND CONTROL 2022; 10:1770-1777. [PMID: 35317431 PMCID: PMC8931598 DOI: 10.1007/s40435-022-00935-3] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 11/27/2021] [Revised: 02/06/2022] [Accepted: 02/24/2022] [Indexed: 11/15/2022]
Abstract
In this paper, we study a diffusive SIR epidemic model with saturated incidence rate and discontinuous treatments under Neumann boundary conditions. Firstly, the existence and boundedness of the solution of the system are addressed. Then, on the basis of the differential inclusions theory, we analysis the existence of endemic equilibrium. Furthermore, by constructing different appropriate Lyapunov functions, we investigate the global asymptotic stability of the disease free equilibrium(DFE) and the endemic equilibrium(EE), respectively. Additionally, numerical simulations are given to confirm the correctness of theorem. Finally, we give a brief conclusion and discussion in the end of the paper.
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Zhao Z, Li Y, Feng Z. Traveling wave phenomena in a nonlocal dispersal predator-prey system with the Beddington-DeAngelis functional response and harvesting. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2021; 18:1629-1652. [PMID: 33757202 DOI: 10.3934/mbe.2021084] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/12/2023]
Abstract
This paper is devoted to studying the existence and nonexistence of traveling wave solution for a nonlocal dispersal delayed predator-prey system with the Beddington-DeAngelis functional response and harvesting. By constructing the suitable upper-lower solutions and applying Schauder's fixed point theorem, we show that there exists a positive constant $c^*$ such that the system possesses a traveling wave solution for any given $c> c^*$. Moreover, the asymptotic behavior of traveling wave solution at infinity is obtained by the contracting rectangles method. The existence of traveling wave solution for $c=c^*$ is established by means of Corduneanu's theorem. The nonexistence of traveling wave solution in the case of $c<c^*$ is also discussed.
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Affiliation(s)
- Zhihong Zhao
- School of Mathematics and Physics, University of Science & Technology Beijing, Beijing 100083, China
| | - Yan Li
- School of Mathematics and Physics, University of Science & Technology Beijing, Beijing 100083, China
| | - Zhaosheng Feng
- School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, Edinburg, Texas 78539, USA
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Roy J, Barman D, Alam S. Role of fear in a predator-prey system with ratio-dependent functional response in deterministic and stochastic environment. Biosystems 2020; 197:104176. [PMID: 32628979 DOI: 10.1016/j.biosystems.2020.104176] [Citation(s) in RCA: 14] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/25/2019] [Revised: 05/13/2020] [Accepted: 05/26/2020] [Indexed: 11/25/2022]
Abstract
In this article, we propose and analyse a predator-prey model where apart from direct predation the prey population is affected by the fear induced from predators. The reproduction of the prey population is reduced as a cost of fear. The predator is assumed to consume the prey according to ratio-dependent functional response and is also involved in intra-specific competition due to limited resources of food. Through model analysis, it has been observed that fear factor regulates the dynamics of the system in a completely different way than in the case where functional response is only prey dependent. Also, intra-specific competition among predators reduces the effect of fear and it forms a different pattern in the system dynamics than that of the effect of fear. Furthermore, the deterministic model has been extended to a stochastic model by perturbing the natural death rates of both prey and predators. It has been observed that the stochastic system possesses a unique positive solution that is globally stable with respect to anywhere in the interior of the positive quadrant. The stochastic extinction and persistence scenario for both the species have been analysed and a detailed comparison between the deterministic and stochastic models have been done through exhaustive numerical simulation. Finally, numerical simulation has been performed to figure out the impact of fear on the population dynamics.
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Affiliation(s)
- Jyotirmoy Roy
- Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, B. Garden, Howrah, 711103, India
| | - Dipesh Barman
- Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, B. Garden, Howrah, 711103, India
| | - Shariful Alam
- Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, B. Garden, Howrah, 711103, India.
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Zhou Z, Van Gorder RA. Turing Instability and Colony Formation in Spatially Extended Rosenzweig-MacArthur Predator-Prey Models with Allochthonous Resources. Bull Math Biol 2019; 81:5009-5053. [PMID: 31595381 DOI: 10.1007/s11538-019-00667-0] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/12/2019] [Accepted: 09/26/2019] [Indexed: 10/25/2022]
Abstract
While it is somewhat well known that spatial PDE extensions of the Rosenzweig-MacArthur predator-prey model do not admit spatial pattern formation through the Turing mechanism, in this paper we demonstrate that the addition of allochthonous resources into the system can result in spatial patterning and colony formation. We study pattern formation, through Turing and Turing-Hopf mechanisms, in two distinct spatial Rosenzweig-MacArthur models generalized to include allochthonous resources. Both models have previously been shown to admit heterogeneous spatial solutions when prey and allochthonous resources are confined to different regions of the domain, with the predator able to move between the regions. However, pattern formation in such cases is not due to the Turing mechanism, but rather due to the spatial separation between the two resources for the predator. On the other hand, for a variety of applications, a predator can forage over a region where more than one food source is present, and this is the case we study in the present paper. We first consider a three PDE model, consisting of equations for each of a predator, a prey, and an allochthonous resource or subsidy, with all three present over the spatial domain. The second model we consider arises in the study of two independent predator-prey systems in which a portion of the prey in the first system becomes an allochthonous resource for the second system; this is referred to as a predator-prey-quarry-resource-scavenger model. We show that there exist parameter regimes for which these systems admit Turing and Turing-Hopf bifurcations, again resulting in spatial or spatiotemporal patterning and hence colony formation. This is interesting from a modeling standpoint, as the standard spatially extended Rosenzweig-MacArthur predator-prey equations do not permit the Turing instability, and hence, the inclusion of allochthonous resources is one route to realizing colony formation under Rosenzweig-MacArthur kinetics. Concerning the ecological application, we find that spatial patterning occurs when the predator is far more mobile than the prey (reflected in the relative difference between their diffusion parameters), with the prey forming colonies and the predators more uniformly dispersed throughout the domain. We discuss how this spatially heterogeneous patterning, particularly of prey populations, may constitute one way in which the paradox of enrichment is resolved in spatial systems by way of introducing allochthonous resource subsidies in conjunction with spatial diffusion of predator and prey populations.
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Affiliation(s)
- Zhi Zhou
- Department of Engineering Sciences and Applied Mathematics, McCormick School of Engineering and Applied Science, Northwestern University, 2145 Sheridan Road, Evanston, IL, 60208, USA
| | - Robert A Van Gorder
- Department of Mathematics and Statistics, University of Otago, P.O. Box 56, Dunedin, 9054, New Zealand.
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Chen D, Wang D. Almost periodic dynamics of delayed prey–predator model with discontinuous harvesting policies and Hassell–Varley type functional response. INT J BIOMATH 2018. [DOI: 10.1142/s1793524518500833] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
In this study, a class of delayed prey–predator model with Hassell–Varley type functional response is investigated, in which the harvesting policies are modeled by discontinuous functions. Based on functional differential inclusions and set-valued analysis theories, the local and global existence of positive solution in sense of Filippov is given. By employing theory of functional differential inclusions and generalized differential inequalities, some sufficient conditions which guarantee the permanence of the system are obtained. According to non-smooth analysis theory with generalized Lyapunov functional approach, a series of useful criteria on existence, uniqueness and global asymptotic stability of the almost positive periodic solution to the system are derived. Some suitable examples together with their numerical simulations are given to illustrate the effectiveness of our results by using MATLAB.
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Affiliation(s)
- Dongxiao Chen
- Fujian Province University Key Laboratory of Computation Science, School of Mathematical Sciences, Huaqiao University, Quanzhou, Fujian 362021, P. R. China
| | - Dongshu Wang
- Fujian Province University Key Laboratory of Computation Science, School of Mathematical Sciences, Huaqiao University, Quanzhou, Fujian 362021, P. R. China
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Self-Organized Patterns Induced by Neimark-Sacker, Flip and Turing Bifurcations in a Discrete Predator-Prey Model with Lesie-Gower Functional Response. ENTROPY 2017. [DOI: 10.3390/e19060258] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
The formation of self-organized patterns in predator-prey models has been a very hot topic recently. The dynamics of these models, bifurcations and pattern formations are so complex that studies are urgently needed. In this research, we transformed a continuous predator-prey model with Lesie-Gower functional response into a discrete model. Fixed points and stability analyses were studied. Around the stable fixed point, bifurcation analyses including: flip, Neimark-Sacker and Turing bifurcation were done and bifurcation conditions were obtained. Based on these bifurcation conditions, parameters values were selected to carry out numerical simulations on pattern formation. The simulation results showed that Neimark-Sacker bifurcation induced spots, spirals and transitional patterns from spots to spirals. Turing bifurcation induced labyrinth patterns and spirals coupled with mosaic patterns, while flip bifurcation induced many irregular complex patterns. Compared with former studies on continuous predator-prey model with Lesie-Gower functional response, our research on the discrete model demonstrated more complex dynamics and varieties of self-organized patterns.
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Luo D, Wang D. Impact of discontinuous harvesting policies on prey–predator system with Hassell–Varley-type functional response. INT J BIOMATH 2017. [DOI: 10.1142/s1793524517500486] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
In this paper, a delayed prey–predator model with discontinuous harvesting policies is investigated, in which the predator population consumes the prey according to Hassell–Varley-type functional response. Under some reasonable assumptions on the discontinuous harvesting functions, the local existence and global existence of positive solution in sense of Filippov to the system are obtained. Based on the functional differential inclusions theory and the topological degree theory of set-valued analysis, a series of useful criteria on existence and non-existence of the positive periodic solution is established for the system. Finally, some numerical examples are given to show the applicability and effectiveness of the obtained results. It is worthy to point out that the discontinuous harvesting policies are superior to continuous harvesting policies, which are usually adopted in previous papers.
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Affiliation(s)
- Daozhong Luo
- School of Mathematical Sciences, Huaqiao University, Quanzhou, Fujian 362021, P. R. China
| | - Dongshu Wang
- School of Mathematical Sciences, Huaqiao University, Quanzhou, Fujian 362021, P. R. China
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Guin LN, Mondal B, Chakravarty S. Stationary patterns induced by self- and cross-diffusion in a Beddington–DeAngelis predator–prey model. ACTA ACUST UNITED AC 2016. [DOI: 10.1007/s40435-016-0281-7] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
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Guin LN, Mondal B, Chakravarty S. Existence of spatiotemporal patterns in the reaction–diffusion predator–prey model incorporating prey refuge. INT J BIOMATH 2016. [DOI: 10.1142/s1793524516500856] [Citation(s) in RCA: 22] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
The pattern formation in reaction–diffusion system has long been the subject of interest to the researchers in the domain of mathematical ecology because of its universal existence and importance. The present investigation deals with a spatial dynamics of the Beddington–DeAngelis predator–prey model in the presence of a constant proportion of prey refuge. The model system representing boundary value problem under study is subjected to homogeneous Neumann boundary conditions. The asymptotic stability including the local and the global stability and the bifurcation as well of the unique positive homogeneous steady state of the corresponding temporal model has been analyzed. The Turing instability region in two-parameter space and the condition of diffusion-driven instability of the spatiotemporal model are investigated. Based on the appropriate numerical simulations, the present model dynamics in Turing space appears to get influenced by prey refuge while it exhibits diffusion-controlled pattern formation growth to spots, stripe-spot mixtures, labyrinthine, stripe-hole mixtures and holes replication. The results obtained appear to enrich the findings of the model system under consideration.
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Affiliation(s)
- Lakshmi Narayan Guin
- Department of Mathematics, Visva-Bharati, Santiniketan-731 235, West Bengal, India
| | - Benukar Mondal
- Department of Mathematics, Visva-Bharati, Santiniketan-731 235, West Bengal, India
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Study of a tri-trophic prey-dependent food chain model of interacting populations. Math Biosci 2013; 246:55-71. [DOI: 10.1016/j.mbs.2013.07.021] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/26/2013] [Revised: 07/08/2013] [Accepted: 07/17/2013] [Indexed: 11/18/2022]
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