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Dey S, Ghorai S, Banerjee M. Analytical detection of stationary and dynamic patterns in a prey-predator model with reproductive Allee effect in prey growth. J Math Biol 2023; 87:21. [PMID: 37395822 DOI: 10.1007/s00285-023-01957-x] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/03/2023] [Revised: 06/06/2023] [Accepted: 06/20/2023] [Indexed: 07/04/2023]
Abstract
Allee effect in population dynamics has a major impact in suppressing the paradox of enrichment through global bifurcation, and it can generate highly complex dynamics. The influence of the reproductive Allee effect, incorporated in the prey's growth rate of a prey-predator model with Beddington-DeAngelis functional response, is investigated here. Preliminary local and global bifurcations are identified of the temporal model. Existence and non-existence of heterogeneous steady-state solutions of the spatio-temporal system are established for suitable ranges of parameter values. The spatio-temporal model satisfies Turing instability conditions, but numerical investigation reveals that the heterogeneous patterns corresponding to unstable Turing eigenmodes act as a transitory pattern. Inclusion of the reproductive Allee effect in the prey population has a destabilising effect on the coexistence equilibrium. For a range of parameter values, various branches of stationary solutions including mode-dependent Turing solutions and localized pattern solutions are identified using numerical bifurcation technique. The model is also capable to produce some complex dynamic patterns such as travelling wave, moving pulse solution, and spatio-temporal chaos for certain range of parameters and diffusivity along with appropriate choice of initial conditions. Judicious choices of parametrization for the Beddington-DeAngelis functional response help us to infer about the resulting patterns for similar prey-predator models with Holling type-II functional response and ratio-dependent functional response.
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Affiliation(s)
- Subrata Dey
- Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, 208016, India
| | - S Ghorai
- Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, 208016, India
| | - Malay Banerjee
- Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, 208016, India.
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He M, Li Z. Stability of a fear effect predator-prey model with mutual interference or group defense. JOURNAL OF BIOLOGICAL DYNAMICS 2022; 16:480-498. [PMID: 35759246 DOI: 10.1080/17513758.2022.2091800] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/28/2021] [Accepted: 06/03/2022] [Indexed: 06/15/2023]
Abstract
In this paper, we consider a fear effect predator-prey model with mutual interference or group defense. For the model with mutual interference, we show the interior equilibrium is globally stable, and the mutual interference can stabilize the predator-prey system. For the model with group defense, we discuss the singular dynamics around the origin and the occurrence of Hopf bifurcation, and find that there is a separatrix curve near the origin such that the orbits above which tend to the origin and the orbits below which tend to limit cycle or the interior equilibrium.
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Affiliation(s)
- Mengxin He
- College of Mathematics and Data Science, Minjiang University, Fuzhou, People's Republic of China
| | - Zhong Li
- School of Mathematics and Statistics, Fuzhou University, Fuzhou, People's Republic of China
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Arroyo-Esquivel J, Hastings A, Baskett ML. Characterizing Long Transients in Consumer–Resource Systems With Group Defense and Discrete Reproductive Pulses. Bull Math Biol 2022; 84:102. [PMID: 35964274 PMCID: PMC9376152 DOI: 10.1007/s11538-022-01059-7] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/31/2022] [Accepted: 07/25/2022] [Indexed: 11/24/2022]
Abstract
AbstractDuring recent years, the study of long transients has been expanded in ecological theory to account for shifts in long-term behavior of ecological systems. These long transients may lead to regime shifts between alternative states that resemble the dynamics of alternative stable states for a prolonged period of time. One dynamic that potentially leads to long transients is the group defense of a resource in a consumer–resource interaction. Furthermore, time lags in the population caused by discrete reproductive pulses have the potential to produce long transients, either independently or in conjunction to the transients caused by the group defense. In this work, we analyze the potential for long transients in a model for a consumer–resource system in which the resource exhibits group defense and reproduces in discrete reproductive pulses. This system exhibits crawl-by transients near the extinction and carrying capacity states of resource, and a transcritical bifurcation, under which a ghost limit cycle appears. We estimate the transient time of our system from these transients using perturbation theory. This work advances an understanding of how systems shift between alternate states and their duration of staying in a given regime and what ecological dynamics may lead to long transients.
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Affiliation(s)
| | - Alan Hastings
- Department of Environmental Science and Policy, University of California, Davis, CA, 95616, USA
| | - Marissa L Baskett
- Department of Environmental Science and Policy, University of California, Davis, CA, 95616, USA
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Optimal harvesting and stability of a predator-prey model for fish populations with schooling behavior. Theory Biosci 2021; 140:225-239. [PMID: 34033110 DOI: 10.1007/s12064-021-00347-5] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/28/2019] [Accepted: 05/04/2021] [Indexed: 10/21/2022]
Abstract
In this paper, the schooling behavior of prey fish population in a predator-prey interaction is investigated. By taking an economical interest which can be elaborated by the presence of nonselective harvesting into consideration, we studied the dynamical behavior. The existence, positivity and boundedness of solution have been established. The analysis of the equilibrium states is presented by studying the local and the global stability. The possible types of local bifurcation that the system can undergoes are discussed. The effect of fishing effort on the evolution of the species is examined. Further, by using Pontryagin's maximum principle a proper management strategy has been used for avoiding the extinction of the considered species and maximizing the benefits. For the validation of the theoretical result, several of graphical representations have been used.
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Djilali S. Spatiotemporal patterns induced by cross-diffusion in predator–prey model with prey herd shape effect. INT J BIOMATH 2020. [DOI: 10.1142/s1793524520500308] [Citation(s) in RCA: 30] [Impact Index Per Article: 7.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
In this paper, we investigate a predator–prey model with herd behavior and cross-diffusion subject to the zero flux boundary conditions. First, the temporal behavior of the model has been investigated, where Hopf bifurcation has been obtained. Then, by analyzing the characteristic equation it has been proved that the cross-diffusion generate a complex dynamics such as Hopf bifurcation, Turing instability, even Turing–Hopf bifurcation. Further, the impact of the prey herd shape on the spatiotemporal patterns has been discussed. Furthermore, by computing and analyzing the normal form associated with the Turing–Hopf bifurcation point, the spatiotemporal dynamics near the Turing–Hopf bifurcation point has been discussed and also justified by some numerical simulations.
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Affiliation(s)
- Salih Djilali
- Faculty of Exact Sciences and Informatics, Mathematics Department, University of Chlef, Laboratoire D’analyse Non-Linéaire et Mathématiques Appliquées, Université de Tlemcen, Tlemcen, Algérie
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CHAKRABORTY PRABIR, GHOSH UTTAM, SARKAR SUSMITA. STABILITY AND BIFURCATION ANALYSIS OF A DISCRETE PREY–PREDATOR MODEL WITH SQUARE-ROOT FUNCTIONAL RESPONSE AND OPTIMAL HARVESTING. J BIOL SYST 2020. [DOI: 10.1142/s0218339020500047] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
In this paper, we have considered a discrete prey–predator model with square-root functional response and optimal harvesting policy. This type of functional response is used to study the dynamics of the prey–predator model where the prey population exhibits herd behavior, i.e., the interaction between prey and predator occurs along the boundary of the population. The considered population model has three fixed points; one is trivial, the second one is axial and the last one is an interior fixed point. The first two fixed points are always feasible but the last one depends on the parameter value. The interior fixed point experiences the flip and Neimark–Sacker bifurcations depending on the predator harvesting coefficient. Finally, an optimal harvesting policy has been introduced and the optimal value of the harvesting coefficient is determined.
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Affiliation(s)
- PRABIR CHAKRABORTY
- Department of Mathematics, Government General Degree College, Muragachha, Muragachha, Nadia 741154, West Bengal, India
| | - UTTAM GHOSH
- Department of Applied Mathematics, University of Calcutta, Kolkata 700009, West Bengal, India
| | - SUSMITA SARKAR
- Department of Applied Mathematics, University of Calcutta, Kolkata 700009, West Bengal, India
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Tang B. Dynamics for a fractional-order predator-prey model with group defense. Sci Rep 2020; 10:4906. [PMID: 32184437 PMCID: PMC7078230 DOI: 10.1038/s41598-020-61468-3] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/17/2019] [Accepted: 02/23/2020] [Indexed: 12/02/2022] Open
Abstract
In the present article, a new fractional order predator-prey model with group defense is put up. The dynamical properties such as the existence, uniqueness and boundness of solution, the stability of equilibrium point and the existence of Hopf bifurcation of the involved predator-prey model have been discussed. Firstly, we establish the sufficient conditions that guarantee the existence, uniqueness and boundness of solution by applying Lipschitz condition, inequality technique and fractional order differential equation theory. Secondly, we analyze the existence of various equilibrium points by basic mathematical analysis method and obtain some sufficient criteria which guarantee the locally asymptotically stability of various equilibrium points of the involved predator-prey model with the aid of linearization approach. Thirdly, the existence of Hopf bifurcation of the considered predator-prey model is investigated by using the Hopf bifurcation theory of fractional order differential equations. Finally, simulation results are presented to substantiate the theoretical findings.
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Affiliation(s)
- Bingnan Tang
- Business school, Jiangsu University of Technology, Changzhou, 213001, P.R. China.
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Choi J, Kim YJ. Predator-prey equations with constant harvesting and planting. J Theor Biol 2018; 458:47-57. [PMID: 30194968 DOI: 10.1016/j.jtbi.2018.08.044] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/29/2018] [Revised: 08/25/2018] [Accepted: 08/29/2018] [Indexed: 11/30/2022]
Abstract
We propose Lotka-Volterra type predator-prey equations which include small constant terms. Depending on its sign, the constant may model various things. To see the effect of the constants clearly we drop all other functional responses except the ones in the original Lotka-Volterra equations. We add a small negative constant for the harvesting or the Allee effect. A positive constant is added to model the planting or external influx. We find the predator-prey equations with constant terms produce most of dynamic and static patterns observed from other predator-prey models with various functional responses.
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Affiliation(s)
- Jieun Choi
- Department of Mathematical Sciences Korea Advanced Institute of Science and Technology, South Korea
| | - Yong-Jung Kim
- Department of Mathematical Sciences Korea Advanced Institute of Science and Technology, South Korea.
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Melchionda D, Pastacaldi E, Perri C, Banerjee M, Venturino E. Social behavior-induced multistability in minimal competitive ecosystems. J Theor Biol 2018; 439:24-38. [DOI: 10.1016/j.jtbi.2017.11.016] [Citation(s) in RCA: 14] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/06/2017] [Revised: 08/25/2017] [Accepted: 11/21/2017] [Indexed: 10/18/2022]
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La Morgia V, Venturino E. Understanding hybridization and competition processes between hare species: Implications for conservation and management on the basis of a mathematical model. Ecol Modell 2017. [DOI: 10.1016/j.ecolmodel.2017.09.002] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
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Ecoepidemic predator-prey model with feeding satiation, prey herd behavior and abandoned infected prey. Math Biosci 2016; 274:58-72. [PMID: 26874217 DOI: 10.1016/j.mbs.2016.02.003] [Citation(s) in RCA: 38] [Impact Index Per Article: 4.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/02/2015] [Revised: 12/23/2015] [Accepted: 02/01/2016] [Indexed: 11/27/2022]
Abstract
In this paper we analyse a predator-prey model where the prey population shows group defense and the prey individuals are affected by a transmissible disease. The resulting model is of the Rosenzweig-MacArthur predator-prey type with an SI (susceptible-infected) disease in the prey. Modeling prey group defense leads to a square root dependence in the Holling type II functional for the predator-prey interaction term. The system dynamics is investigated using simulations, classical existence and asymptotic stability analysis and numerical bifurcation analysis. A number of bifurcations, such as transcritical and Hopf bifurcations which occur commonly in predator-prey systems will be found. Because of the square root interaction term there is non-uniqueness of the solution and a singularity where the prey population goes extinct in a finite time. This results in a collapse initiated by extinction of the healthy or susceptible prey and thereafter the other population(s). When also a positive attractor exists this leads to bistability similar to what is found in predator-prey models with a strong Allee effect. For the two-dimensional disease-free (i.e. the purely demographic) system the region in the parameter space where bistability occurs is marked by a global bifurcation. At this bifurcation a heteroclinic connection exists between saddle prey-only equilibrium points where a stable limit cycle together with its basin of attraction, are destructed. In a companion paper (Gimmelli et al., 2015) the same model was formulated and analysed in which the disease was not in the prey but in the predator. There we also observed this phenomenon. Here we extend its analysis using a phase portrait analysis. For the three-dimensional ecoepidemic predator-prey system where the prey is affected by the disease, also tangent bifurcations including a cusp bifurcation and a torus bifurcation of limit cycles occur. This leads to new complex dynamics. Continuation by varying one parameter of the emerging quasi-periodic dynamics from a torus bifurcation can lead to its destruction by a collision with a saddle-cycle. Under other conditions the quasi-periodic dynamics changes gradually in a trajectory that lands on a boundary point where the prey go extinct in finite time after which a total collapse of the three-dimensional system occurs.
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Gimmelli G, Kooi BW, Venturino E. Ecoepidemic models with prey group defense and feeding saturation. ECOLOGICAL COMPLEXITY 2015. [DOI: 10.1016/j.ecocom.2015.02.004] [Citation(s) in RCA: 35] [Impact Index Per Article: 3.9] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 10/23/2022]
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