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Spatiotemporal Dynamics in a Predator–Prey Model with Functional Response Increasing in Both Predator and Prey Densities. MATHEMATICS 2021. [DOI: 10.3390/math10010017] [Citation(s) in RCA: 7] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
In this paper, a diffusive predator–prey system with a functional response that increases in both predator and prey densities is considered. By analyzing the characteristic roots of the partial differential equation system, the Turing instability and Hopf bifurcation are studied. In order to consider the dynamics of the model where the Turing bifurcation curve and the Hopf bifurcation curve intersect, we chose the diffusion coefficients d1 and β as bifurcating parameters. In particular, the normal form of Turing–Hopf bifurcation was calculated so that we could obtain the phase diagram. For parameters in each region of the phase diagram, there are different types of solutions, and their dynamic properties are extremely rich. In this study, we have used some numerical simulations in order to confirm these ideas.
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Yang Y, Liu LR, Xiang CC, Qin WJ. Switching dynamics analysis of forest-pest model describing effects of external periodic disturbance. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2020; 17:4328-4347. [PMID: 32987582 DOI: 10.3934/mbe.2020239] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/11/2023]
Abstract
A periodically forced Filippov forest-pest model incorporating threshold policy control and integrated pest management is proposed. It is very natural and reasonable to introduce Filippov non-smooth system into the ecosystem since there were many disadvantageous factors in pest control at fixed time and the threshold control according to state variable showed rewarding characteristics. The main aim of this paper is to quest the association between pests dynamics and system parameters especially the economical threshold ET, the amplitude and frequency of periodic forcing term. From the view of pest control, if the maximum amplitude of the sliding periodic solution does not exceed economic injury level(EIL), the sliding periodic solution is a desired result for pest control. The Filippov forest-pest model exhibits the rich dynamic behaviors including multiple attractors coexistence, period-adding bifurcation, quasi-periodic feature and chaos. At certain frequency of periodic forcing, the varying system initial densities trigger the system state switch between different attractors with diverse amplitudes and periods. Besides, parameters sensitivity analysis shows that the pest could be controlled at a certain level by choosing suitable parameters.
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Affiliation(s)
- Yi Yang
- College of Computer Science and Engineering, Chongqing Three Gorges University, Chongqing 404100, China
- Key Laboratory of Intelligent Information Processing and Control of Chongqing Municipal Institutions of Higher education, Chongqing Three Gorges University, Chongqing 404100, China
| | - Li Rong Liu
- School of Mathematics and Statistics, Hubei Minzu University, Enshi 445000, China
| | - Chang Cheng Xiang
- School of Mathematics and Statistics, Hubei Minzu University, Enshi 445000, China
| | - Wen Jie Qin
- Three Gorges Mathematical Research Center, China Three Gorges University, Yichang 443002, China
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Sauve AMC, Taylor RA, Barraquand F. The effect of seasonal strength and abruptness on predator-prey dynamics. J Theor Biol 2020; 491:110175. [PMID: 32017869 DOI: 10.1016/j.jtbi.2020.110175] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/26/2019] [Revised: 01/05/2020] [Accepted: 01/24/2020] [Indexed: 10/25/2022]
Abstract
Coupled dynamical systems in ecology are known to respond to the seasonal forcing of their parameters with multiple dynamical behaviours, ranging from seasonal cycles to chaos. Seasonal forcing is predominantly modelled as a sine wave. However, the transition between seasons is often more sudden as illustrated by the effect of snow cover on predation success. A handful of studies have mentioned the robustness of their results to the shape of the forcing signal but did not report any detailed analyses. Therefore, whether and how the shape of seasonal forcing could affect the dynamics of coupled dynamical systems remains unclear, while abrupt seasonal transitions are widespread in ecological systems. To provide some answers, we conduct a numerical analysis of the dynamical response of predator-prey communities to the shape of the forcing signal by exploring the joint effect of two features of seasonal forcing: the magnitude of the signal, which is classically the only one studied, and the shape of the signal, abrupt or sinusoidal. We consider both linear and saturating functional responses, and focus on seasonal forcing of the predator's discovery rate, which fluctuates with changing environmental conditions and prey's ability to escape predation. Our numerical results highlight that a more abrupt seasonal forcing mostly alters the magnitude of population fluctuations and triggers period-doubling bifurcations, as well as the emergence of chaos, at lower forcing strength than for sine waves. Controlling the variance of the forcing signal mitigates this trend but does not fully suppress it, which suggests that the variance is not the only feature of the shape of seasonal forcing that acts on community dynamics. Although theoretical studies may predict correctly the sequence of bifurcations using sine waves as a representation of seasonality, there is a rationale for applied studies to implement as realistic seasonal forcing as possible to make precise predictions of community dynamics.
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Affiliation(s)
- Alix M C Sauve
- University of Bordeaux, Integrative and Theoretical Ecology, LabEx COTE, France.
| | - Rachel A Taylor
- Animal and Plant Health Agency (APHA), Weybridge, United Kingdom
| | - Frédéric Barraquand
- University of Bordeaux, Integrative and Theoretical Ecology, LabEx COTE, France; CNRS, Institute of Mathematics of Bordeaux, France
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Tzuk O, Ujjwal SR, Fernandez-Oto C, Seifan M, Meron E. Period doubling as an indicator for ecosystem sensitivity to climate extremes. Sci Rep 2019; 9:19577. [PMID: 31862940 PMCID: PMC6925204 DOI: 10.1038/s41598-019-56080-z] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/17/2019] [Accepted: 11/13/2019] [Indexed: 11/21/2022] Open
Abstract
The predictions for a warmer and drier climate and for increased likelihood of climate extremes raise high concerns about the possible collapse of dryland ecosystems, and about the formation of new drylands where native species are less tolerant to water stress. Using a dryland-vegetation model for plant species that display different tradeoffs between fast growth and tolerance to droughts, we find that ecosystems subjected to strong seasonal variability, typical for drylands, exhibit a temporal period-doubling route to chaos that results in early collapse to bare soil. We further find that fast-growing plants go through period doubling sooner and span wider chaotic ranges than stress-tolerant plants. We propose the detection of period-doubling signatures in power spectra as early indicators of ecosystem collapse that outperform existing indicators in their ability to warn against climate extremes and capture the heightened vulnerability of newly-formed drylands. The proposed indicator is expected to apply to other types of ecosystems, such as consumer–resource and predator–prey systems. We conclude by delineating the conditions ecosystems should meet in order for the proposed indicator to apply.
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Affiliation(s)
- Omer Tzuk
- Department of Physics, Ben-Gurion University of the Negev, Beer-Sheva, 8410501, Israel.
| | - Sangeeta Rani Ujjwal
- Department of Solar Energy and Environmental Physics, SIDEER, Blaustein Institutes for Desert Research, Ben-Gurion University of the Negev, Sede Boqer Campus, Midreshet Ben-Gurion, 8499000, Israel
| | - Cristian Fernandez-Oto
- Department of Solar Energy and Environmental Physics, SIDEER, Blaustein Institutes for Desert Research, Ben-Gurion University of the Negev, Sede Boqer Campus, Midreshet Ben-Gurion, 8499000, Israel.,Complex Systems Group, Facultad de Ingeniería y Ciencias Aplicadas, Universidad de los Andes, Av. Mon. Alvaro del Portillo 12.455, Santiago, Chile
| | - Merav Seifan
- Mitrani Department of Desert Ecology, SIDEER, Blaustein Institutes for Desert Research, Ben-Gurion University of the Negev, Sede Boqer Campus, Midreshet Ben-Gurion, 8499000, Israel
| | - Ehud Meron
- Department of Physics, Ben-Gurion University of the Negev, Beer-Sheva, 8410501, Israel.,Department of Solar Energy and Environmental Physics, SIDEER, Blaustein Institutes for Desert Research, Ben-Gurion University of the Negev, Sede Boqer Campus, Midreshet Ben-Gurion, 8499000, Israel
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Haugaard AM. Predicting the bounds of large chaotic systems using low-dimensional manifolds. PLoS One 2017; 12:e0179507. [PMID: 28644871 PMCID: PMC5482462 DOI: 10.1371/journal.pone.0179507] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/03/2016] [Accepted: 05/30/2017] [Indexed: 11/18/2022] Open
Abstract
Predicting extrema of chaotic systems in high-dimensional phase space remains a challenge. Methods, which give extrema that are valid in the long term, have thus far been restricted to models of only a few variables. Here, a method is presented which treats extrema of chaotic systems as belonging to discretised manifolds of low dimension (low-D) embedded in high-dimensional (high-D) phase space. As a central feature, the method exploits that strange attractor dimension is generally much smaller than parent system phase space dimension. This is important, since the computational cost associated with discretised manifolds depends exponentially on their dimension. Thus, systems that would otherwise be associated with tremendous computational challenges, can be tackled on a laptop. As a test, bounding manifolds are calculated for high-D modifications of the canonical Duffing system. Parameters can be set such that the bounding manifold displays harmonic behaviour even if the underlying system is chaotic. Thus, solving for one post-transient forcing cycle of the bounding manifold predicts the extrema of the underlying chaotic problem indefinitely.
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Affiliation(s)
- Asger M. Haugaard
- University of Oxford, Medical sciences division, Oxford, OX3 9DU, United Kingdom
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GAKKHAR SUNITA, SINGH ANURAJ, SINGH BRAHAMPAL. EFFECTS OF DELAY AND SEASONALITY ON TOXIN PRODUCING PHYTOPLANKTON–ZOOPLANKTON SYSTEM. INT J BIOMATH 2012. [DOI: 10.1142/s1793524511001891] [Citation(s) in RCA: 21] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
A dynamical model for toxin producing phytoplankton and zooplankton has been formulated and analyzed. Due to gestation of prey, a discrete time delay is incorporated in the predator dynamics. The stability of the delay model is discussed and Hopf bifurcation to a periodic orbit is established. Stability and direction of bifurcating periodic orbits are investigated using normal form theory and center manifold arguments. Global existence of periodic orbits is also established. To substantiate analytical findings, numerical simulations are performed. The system shows rich dynamic behavior including chaos and limit cycles. The influence of seasonality in intrinsic growth parameter of the phytoplankton population is also investigated. Seasonality leads to complexity in the system.
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Affiliation(s)
- SUNITA GAKKHAR
- Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, Uttarakhand, India
| | - ANURAJ SINGH
- Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, Uttarakhand, India
| | - BRAHAM PAL SINGH
- Department of Mathematics, J. V. Jain Degree College, Saharanpur 247001, U.P., India
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ZHANG YUJUAN, XU ZHILONG, LIU BING, CHEN LANSUN. DYNAMIC ANALYSIS OF A HOLLING I PREDATOR-PREY SYSTEM WITH MUTUAL INTERFERENCE CONCERNING PEST CONTROL. J BIOL SYST 2011. [DOI: 10.1142/s0218339005001392] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
A Holling I predator-prey model with mutual interference concerning pest control is proposed and analyzed. The prey and predator are considered to be a pest and a natural enemy, respectively. The model is forced by the addition of periodic impulsive terms representing predator import (biological control) and pesticide application (chemical control) at different fixed moments. By using Floquet theory and small amplitude perturbations, we show the existence and stability of pest-free periodic solutions. Further, we prove that when the stability of pest-free periodic solutions is lost, the system is permanent by using analytic methods of differential equation theory. Numerical solutions are also given, which show that when stability of pest-free periodic solutions is lost, more exotic behavior can occur, such as quasi-periodic oscillation or chaos. We investigate the effect of impulsive perturbations on the unforced continuous system, and find that the forced system has a different dynamical behavior with a different range of initial values which are inside or outside the unstable limit cycle of the unforced continuous system. Finally, we compare the validity of the combination of biological control and chemical control with classical methods and conclude that the synthetical strategy is more effective than classical methods if we take effective chemical control.
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Affiliation(s)
- YUJUAN ZHANG
- Department of Mathematics, Anshan Normal University, Anshan, Liaoning 114005, P. R. China
- Department of Bioscience and Biotechnology, Dalian University of Technology, Dalian, Liaoning 116024, P. R. China
| | - ZHILONG XU
- Department of Bioscience and Biotechnology, Dalian University of Technology, Dalian, Liaoning 116024, P. R. China
| | - BING LIU
- Department of Mathematics, Anshan Normal University, Anshan, Liaoning 114005, P. R. China
| | - LANSUN CHEN
- Institute of Mathematics, Academy of Mathematics and System Sciences, Academia Sinica, Beijing 100080, P. R. China
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Permanence and stability of an Ivlev-type predator–prey system with impulsive control strategies. ACTA ACUST UNITED AC 2009. [DOI: 10.1016/j.mcm.2009.07.007] [Citation(s) in RCA: 27] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/20/2022]
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Baek H. Species extinction and permanence of an impulsively controlled two-prey one-predator system with seasonal effects. Biosystems 2009; 98:7-18. [PMID: 19591895 DOI: 10.1016/j.biosystems.2009.06.008] [Citation(s) in RCA: 22] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/05/2009] [Revised: 04/21/2009] [Accepted: 06/30/2009] [Indexed: 11/19/2022]
Abstract
Recently, the population dynamic systems with impulsive controls have been researched by many authors. However, most of them are reluctant to study the seasonal effects on prey. Thus, in this paper, an impulsively controlled two-prey one-predator system with the Beddington-DeAngelis type functional response and seasonal effects is investigated. By using the Floquet theory, the sufficient conditions for the existence of a globally asymptotically stable two-prey-free periodic solution are established. Further, it is proven that this system is permanent under some conditions via a comparison method involving multiple Lyapunov functions and meanwhile the conditions for extinction of one of the two prey and permanence of the remaining two species are given.
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Affiliation(s)
- Hunki Baek
- Department of Mathematics, Kyungpook National University, Sankyeuk Dong, Daegu 702-701, South Korea.
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Castilho C, Srinivasu PDN. Bio-economics of a renewable resource in a seasonally varying environment. Math Biosci 2007; 205:1-18. [PMID: 17087978 DOI: 10.1016/j.mbs.2006.09.011] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/26/2004] [Revised: 04/21/2005] [Accepted: 09/12/2006] [Indexed: 11/17/2022]
Abstract
In this paper we study the bio-economics of a renewable resource with governing dynamics described by two distinct growth functions (viz., logistic and Gompertz growth functions) in a seasonally varying environment. Seasonality is introduced into the system by taking the involved ecological parameters to be periodic. In this work, we establish a procedure to obtain the optimal path and compute the optimal effort policy which maximizes the net revenue to the harvester for a fairly general optimal control problem and apply this procedure to the considered models to derive some important conclusions. These problems are solved on the infinite horizon. We find that, for both the models, the optimal harvest policy and the corresponding optimal path are periodic after a finite time. We also obtain optimal solution, a suboptimal harvesting policy and the corresponding suboptimal approach path to reach this optimal solution. The key results are illustrated using numerical simulations and we compare the revenues to the harvester along the optimal and suboptimal paths. The general procedure developed in this work, for obtaining the optimal effort policy and the optimal path, has wider applicability.
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Affiliation(s)
- César Castilho
- Departamento de Matemática, Universidade Federal de Pernambuco, Recife PE, CEP 50740-540, Brazil
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Downs TJ, Ambrose RF. Syntropic Ecotoxicology: A Heuristic Model for Understanding the Vulnerability of Ecological Systems to Stress. ACTA ACUST UNITED AC 2002. [DOI: 10.1046/j.1526-0992.2001.01038.x] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/20/2022]
Affiliation(s)
- Timothy J. Downs
- Environmental Science and Policy Program, Department of International Development, Community and Environment, Clark University, Worcester, Massachusetts
| | - Richard F. Ambrose
- Environmental Science and Engineering Program, School of Public Health, University of California at Los Angeles, Los Angeles, California
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Abstract
A plankton-fish model, comprising phosphorus, algae, zooplankton, and young fish, with light intensity and water temperature varying periodically with the seasons, is analyzed in this paper. For realistic values of the parameters the model behaves chaotically, but its dynamics within the strange attractor can be described by a few one-dimensional maps that allow one to forecast the next yearly peak of plankton or fish from the last peaks. This property is an unambiguous mark of a special form of chaos. Unfortunately, the estimate of such peak-to-peak maps from field data is possible only if plankton or young fish biomass has been sampled accurately and frequently for a paramount number of years. In conclusion, the analysis shows that it might be that plankton dynamics are characterized by an interesting and peculiar form of chaos, but that inferences from recorded data on the existence of these forms of chaos are premature.
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Affiliation(s)
- S Rinaldi
- CIRITA, Politecnico di Milano, Milan, I-20133, Italy.
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