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Enatsu Y, Roy J, Banerjee M. Hunting cooperation in a prey-predator model with maturation delay. JOURNAL OF BIOLOGICAL DYNAMICS 2024; 18:2332279. [PMID: 38517146 DOI: 10.1080/17513758.2024.2332279] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/25/2023] [Accepted: 03/12/2024] [Indexed: 03/23/2024]
Abstract
We investigate the dynamics of a prey-predator model with cooperative hunting among specialist predators and maturation delay in predator growth. First, we consider a model without delay and explore the effect of hunting time on the coexistence of predator and their prey. When the hunting time is long enough and the cooperation rate among predators is weak, prey and predator species tend to coexist. Furthermore, we observe the occurrences of a series of bifurcations that depend on the cooperation rate and the hunting time. Second, we introduce a maturation delay for predator growth and analyse its impact on the system's dynamics. We find that as the delay becomes larger, predator species become more likely to go extinct, as the long maturation delay hinders the growth of the predator population. Our numerical exploration reveals that the delay causes shifts in both the bifurcation curves and bifurcation thresholds of the non-delayed system.
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Affiliation(s)
- Yoichi Enatsu
- Institute of Arts and Sciences, Oshamambe Division, Tokyo University of Science, Hokkaido, Japan
| | - Jyotirmoy Roy
- Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, India
- Department of Basic Science and Humanities, University of Engineering and Management, Kolkata, India
| | - Malay Banerjee
- Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, India
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2
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Wu SX, Wang ZC, Ruan S. Hopf bifurcation in an age-structured predator-prey system with Beddington-DeAngelis functional response and constant harvesting. J Math Biol 2024; 88:56. [PMID: 38573424 PMCID: PMC10995042 DOI: 10.1007/s00285-024-02070-3] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/22/2023] [Revised: 11/26/2023] [Accepted: 02/29/2024] [Indexed: 04/05/2024]
Abstract
In this paper, an age-structured predator-prey system with Beddington-DeAngelis (B-D) type functional response, prey refuge and harvesting is investigated, where the predator fertility function f(a) and the maturation function β ( a ) are assumed to be piecewise functions related to their maturation period τ . Firstly, we rewrite the original system as a non-densely defined abstract Cauchy problem and show the existence of solutions. In particular, we discuss the existence and uniqueness of a positive equilibrium of the system. Secondly, we consider the maturation period τ as a bifurcation parameter and show the existence of Hopf bifurcation at the positive equilibrium by applying the integrated semigroup theory and Hopf bifurcation theorem. Moreover, the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are studied by applying the center manifold theorem and normal form theory. Finally, some numerical simulations are given to illustrate of the theoretical results and a brief discussion is presented.
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Affiliation(s)
- San-Xing Wu
- School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, Gansu, People's Republic of China
| | - Zhi-Cheng Wang
- School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, Gansu, People's Republic of China
| | - Shigui Ruan
- Department of Mathematics, University of Miami, Coral Gables, FL, 33146, USA.
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3
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Chen M, Xu Y, Zhao J, Wei X. Turing-Hopf Bifurcation Analysis in a Diffusive Ratio-Dependent Predator-Prey Model with Allee Effect and Predator Harvesting. ENTROPY (BASEL, SWITZERLAND) 2023; 26:18. [PMID: 38248144 PMCID: PMC10814816 DOI: 10.3390/e26010018] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/22/2023] [Revised: 12/17/2023] [Accepted: 12/20/2023] [Indexed: 01/23/2024]
Abstract
This paper investigates the complex dynamics of a ratio-dependent predator-prey model incorporating the Allee effect in prey and predator harvesting. To explore the joint effect of the harvesting effort and diffusion on the dynamics of the system, we perform the following analyses: (a) The stability of non-negative constant steady states; (b) The sufficient conditions for the occurrence of a Hopf bifurcation, Turing bifurcation, and Turing-Hopf bifurcation; (c) The derivation of the normal form near the Turing-Hopf singularity. Moreover, we provide numerical simulations to illustrate the theoretical results. The results demonstrate that the small change in harvesting effort and the ratio of the diffusion coefficients will destabilize the constant steady states and lead to the complex spatiotemporal behaviors, including homogeneous and inhomogeneous periodic solutions and nonconstant steady states. Moreover, the numerical simulations coincide with our theoretical results.
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Affiliation(s)
| | | | | | - Xin Wei
- Ecological Restoration and Resource Utilization for Cold Region, School of Mathematical Science, Heilongjiang University, Harbin 150080, China; (M.C.); (Y.X.); (J.Z.)
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4
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Liu F, Du Y. Spatiotemporal dynamics of a diffusive predator-prey model with delay and Allee effect in predator. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2023; 20:19372-19400. [PMID: 38052605 DOI: 10.3934/mbe.2023857] [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: 12/07/2023]
Abstract
It has been shown that Allee effect can change predator-prey dynamics and impact species persistence. Allee effect in the prey population has been widely investigated. However, the study on the Allee effect in the predator population is rare. In this paper, we investigate the spatiotemporal dynamics of a diffusive predator-prey model with digestion delay and Allee effect in the predator population. The conditions of stability and instability induced by diffusion for the positive equilibrium are obtained. The effect of delay on the dynamics of system has three different cases: (a) the delay doesn't change the stability of the positive equilibrium, (b) destabilizes and stabilizes the positive equilibrium and induces stability switches, or (c) destabilizes the positive equilibrium and induces Hopf bifurcation, which is revealed (numerically) to be corresponding to high, intermediate or low level of Allee effect, respectively. To figure out the joint effect of delay and diffusion, we carry out Turing-Hopf bifurcation analysis and derive its normal form, from which we can obtain the classification of dynamics near Turing-Hopf bifurcation point. Complex spatiotemporal dynamical behaviors are found, including the coexistence of two stable spatially homogeneous or inhomogeneous periodic solutions and two stable spatially inhomogeneous quasi-periodic solutions. It deepens our understanding of the effects of Allee effect in the predator population and presents new phenomena induced be delay with spatial diffusion.
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Affiliation(s)
- Fang Liu
- School of Mathematics and Data Science, Shaanxi University of Science and Technology, Xi'an 710021, China
| | - Yanfei Du
- School of Mathematics and Data Science, Shaanxi University of Science and Technology, Xi'an 710021, China
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5
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Tripathi JP, Tripathi D, Mandal S, Shrimali MD. Cannibalistic enemy-pest model: effect of additional food and harvesting. J Math Biol 2023; 87:58. [PMID: 37702756 DOI: 10.1007/s00285-023-01991-9] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/16/2022] [Revised: 07/28/2023] [Accepted: 08/19/2023] [Indexed: 09/14/2023]
Abstract
Biological control using natural enemies with additional food resources is one of the most adopted and ecofriendly pest control techniques. Moreover, additional food is also provided to natural enemies to divert them from cannibalism. In the present work, using the theory of dynamical system, we discuss the dynamics of a cannibalistic predator prey model in the presence of different harvesting schemes in prey (pest) population and provision of additional food to predators (natural enemies). A detailed mathematical analysis and numerical evaluations have been presented to discuss the pest free state, coexistence of species, stability, occurrence of different bifurcations (saddle-node, transcritical, Hopf, Bogdanov-Takens) and the impact of additional food and harvesting schemes on the dynamics of the system. It has been obtained that the multiple coexisting equilibria and their stability depend on the additional food (quality and quantity) and harvesting rates. Interestingly, we also observe that the pest population density decreases immediately even when small amount of harvesting is implemented. Also the eradication of pest population (stable pest free state) could be achieved via variation in the additional food and implemented harvesting schemes. The individual effects of harvesting parameters on the pest density suggest that the linear harvesting scheme is more effective to control the pest population rather than constant and nonlinear harvesting schemes. In the context of biological control programs, the present theoretical work suggests different threshold values of implemented harvesting and appropriate choices of additional food to be supplied for pest eradication.
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Affiliation(s)
- Jai Prakash Tripathi
- Department of Mathematics, Central University of Rajasthan, NH 8, Bandarsindri, Ajmer, Rajasthan, 305817, India
| | - Deepak Tripathi
- Department of Mathematics, Central University of Rajasthan, NH 8, Bandarsindri, Ajmer, Rajasthan, 305817, India
| | - Swarnendu Mandal
- Department of Physics, Central University of Rajasthan, NH 8, Bandarsindri, Ajmer, Rajasthan, 305817, India
| | - Manish Dev Shrimali
- Department of Physics, Central University of Rajasthan, NH 8, Bandarsindri, Ajmer, Rajasthan, 305817, India.
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6
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Shao Y. Bifurcations of a delayed predator-prey system with fear, refuge for prey and additional food for predator. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2023; 20:7429-7452. [PMID: 37161158 DOI: 10.3934/mbe.2023322] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/11/2023]
Abstract
Taking into account the impacts of the fear by predator, anti-predation response, refuge for prey, additional food supplement for predator and the delayed fear induced by the predator, we establish a delayed predator-prey model in this paper. We analyze the persistence and extinction of species and the existence and uniqueness of a coexistence fixed point. Particularly, we investigate the local asymptotic stability of the equilibrium by use of the characteristic equation theory of a variational matrix. Applying the Hopf bifurcation theorem, we investigate and obtain the bifurcation thresholds of the parameters of fear, refuge coefficient, the quality and quantity of additional food and the anti-predation delayed response produced by prey. Finally we give some examples to verify our theoretical findings and clarify the detailed influences of these parameters on the system dynamics. The main conclusions reveal that these parameters play an important role in the long-term behaviors of species and should be applied correctly to preserve the continuous development of species.
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Affiliation(s)
- Yuanfu Shao
- College of Science, Guilin University of Technology, Guilin, Guangxi 541004, China
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7
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Adan MMIY, Kamrujjaman M, Molla MM, Mohebujjaman M, Buenrostro C. Interplay of harvesting and the growth rate for spatially diversified populations and the testing of a decoupled scheme. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2023; 20:6374-6399. [PMID: 37161112 DOI: 10.3934/mbe.2023276] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/11/2023]
Abstract
The loss and degradation of habitat, Allee effects, climate change, deforestation, hunting-overfishing and human disturbances are alarming and significant threats to the extinction of many species in ecology. When populations compete for natural resources, food supply and habitat, survival to extinction and various other issues are visible. This paper investigates the competition of two species in a heterogeneous environment that are subject to the effect of harvesting. The most realistic harvesting case is connected with the intrinsic growth rate, and the harvesting functions are developed based on this clause instead of random choice. We prove the existence and uniqueness of the solution to the model. Theoretically, we state that, when species coexist, one may drive the other to die out, so both species become extinct, considering all possible rational values of parameters. These results highlight a worthy-of attention study between two populations based on harvesting coefficients. Finally, we solve the model for two spatial dimensions by using a backward Euler, decoupled and linearized time-stepping fully discrete algorithm in a series of examples and observe a match between the theoretical and numerical findings.
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Affiliation(s)
| | - Md Kamrujjaman
- Department of Mathematics, University of Dhaka, Dhaka 1000, Bangladesh
| | - Md Mamun Molla
- Department of Mathematics & Physics, North South University, Dhaka 1229, Bangladesh
- Center for Applied Scientific Computing, North South University, Dhaka 1229, Bangladesh
| | - Muhammad Mohebujjaman
- Department of Mathematics and Physics, Texas A & M International University, Laredo, TX 78041, USA
| | - Clarisa Buenrostro
- Department of Mathematics and Physics, Texas A & M International University, Laredo, TX 78041, USA
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8
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Zhu H, Ji X, Lu J. Impulsive strategies in nonlinear dynamical systems: A brief overview. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2023; 20:4274-4321. [PMID: 36899627 DOI: 10.3934/mbe.2023200] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/18/2023]
Abstract
The studies of impulsive dynamical systems have been thoroughly explored, and extensive publications have been made available. This study is mainly in the framework of continuous-time systems and aims to give an exhaustive review of several main kinds of impulsive strategies with different structures. Particularly, (i) two kinds of impulse-delay structures are discussed respectively according to the different parts where the time delay exists, and some potential effects of time delay in stability analysis are emphasized. (ii) The event-based impulsive control strategies are systematically introduced in the light of several novel event-triggered mechanisms determining the impulsive time sequences. (iii) The hybrid effects of impulses are emphatically stressed for nonlinear dynamical systems, and the constraint relationships between different impulses are revealed. (iv) The recent applications of impulses in the synchronization problem of dynamical networks are investigated. Based on the above several points, we make a detailed introduction for impulsive dynamical systems, and some significant stability results have been presented. Finally, several challenges are suggested for future works.
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Affiliation(s)
- Haitao Zhu
- Department of Systems Science, School of Mathematics, Southeast University, Nanjing 210096, China
| | - Xinrui Ji
- Department of Systems Science, School of Mathematics, Southeast University, Nanjing 210096, China
- The Institute of Complex Networks and Intelligent Systems, Shanghai Research Institute for Intelligent Autonomous Systems, Tongji University, Shanghai 201210, China
| | - Jianquan Lu
- Department of Systems Science, School of Mathematics, Southeast University, Nanjing 210096, China
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9
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Spatiotemporal patterns of a diffusive prey-predator model with spatial memory and pregnancy period in an intimidatory environment. J Math Biol 2022; 84:12. [DOI: 10.1007/s00285-022-01716-4] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/08/2020] [Revised: 09/11/2021] [Accepted: 01/05/2022] [Indexed: 12/17/2022]
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10
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Dynamics in a Predator–Prey Model with Cooperative Hunting and Allee Effect. MATHEMATICS 2021. [DOI: 10.3390/math9243193] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
This paper deals with a diffusive predator–prey model with two delays. First, we consider the local bifurcation and global dynamical behavior of the kinetic system, which is a predator–prey model with cooperative hunting and Allee effect. For the model with weak cooperation, we prove the existence of limit cycle, and a loop of heteroclinic orbits connecting two equilibria at a threshold of conversion rate p=p#, by investigating stable and unstable manifolds of saddles. When p>p#, both species go extinct, and when p<p#, there is a separatrix. The species with initial population above the separatrix finally become extinct, and the species with initial population below it can be coexisting, oscillating sustainably, or surviving of the prey only. In the case with strong cooperation, we exhibit the complex dynamics of system, including limit cycle, loop of heteroclinic orbits among three equilibria, and homoclinic cycle with the aid of theoretical analysis or numerical simulation. There may be three stable states coexisting: extinction state, coexistence or sustained oscillation, and the survival of the prey only, and the attraction basin of each state is obtained in the phase plane. Moreover, we find diffusion may induce Turing instability and Turing–Hopf bifurcation, leaving the system with spatially inhomogeneous distribution of the species, coexistence of two different spatial-temporal oscillations. Finally, we consider Hopf and double Hopf bifurcations of the diffusive system induced by two delays: mature delay of the prey and gestation delay of the predator. Normal form analysis indicates that two spatially homogeneous periodic oscillations may coexist by increasing both delays.
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11
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Lalu M, Phaneendra K, Emineni SP. Numerical approach for differential-difference equations having layer behaviour with small or large delay using non-polynomial spline. Soft comput 2021. [DOI: 10.1007/s00500-021-06032-5] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/20/2022]
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12
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Rihan FA, Alsakaji HJ. Dynamics of a stochastic delay differential model for COVID-19 infection with asymptomatic infected and interacting people: Case study in the UAE. RESULTS IN PHYSICS 2021; 28:104658. [PMID: 34401225 PMCID: PMC8354795 DOI: 10.1016/j.rinp.2021.104658] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/07/2021] [Revised: 08/05/2021] [Accepted: 08/05/2021] [Indexed: 05/12/2023]
Abstract
Public health science is increasingly focusing on understanding how COVID-19 spreads among humans. For the dynamics of COVID-19, we propose a stochastic epidemic model, with time-delays, Susceptible-Infected-Asymptomatic-Quarantined-Recovered (SIAQR). One global positive solution exists with probability one in the model. As a threshold condition of persistence and existence of an ergodic stationary distribution, we deduce a generalized stochastic thresholdR 0 s < R 0 . To estimate the percentages of people who must be vaccinated to achieve herd immunity, least-squares approaches were used to estimateR 0 from real observations in the UAE. Our results suggest that whenR 0 > 1 , a proportion max ( 1 - 1 / R 0 ) of the population needs to be immunized/vaccinated during the pandemic wave. Numerical simulations show that the proposed stochastic delay differential model is consistent with the physical sensitivity and fluctuation of the real observations.
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Affiliation(s)
- F A Rihan
- Department of Mathematical Sciences, College of Science, UAE University, Al-Ain, 15551, United Arab Emirates
| | - H J Alsakaji
- Department of Mathematical Sciences, College of Science, UAE University, Al-Ain, 15551, United Arab Emirates
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13
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Luo Y, Zhang L, Teng Z, Zheng T. Stability and bifurcation for a stochastic differential algebraic Holling-II predator–prey model with nonlinear harvesting and delay. INT J BIOMATH 2021. [DOI: 10.1142/s1793524521500194] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
In this paper, a stochastic delayed differential algebraic predator–prey model with Michaelis–Menten-type prey harvesting is proposed. Due to the influence of gestation delay and stochastic fluctuations, the proposed model displays a complex dynamics. Criteria on the local stability of the interior equilibrium are established, and the effect of gestation delay on the model dynamics is discussed. Taking the gestation delay and economic profit as bifurcation parameters, Hopf bifurcation and singularity induced bifurcation can occur as they cross through some critical values, respectively. Moreover, the solution of the model will blow up in a limited time when delay [Formula: see text]. Then, we calculate the fluctuation intensity of the stochastic fluctuations by Fourier transform method, which is the key to illustrate the effect of stochastic fluctuations. Finally, we demonstrate our theoretical results by numerical simulations.
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Affiliation(s)
- Yantao Luo
- College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, P. R. China
| | - Long Zhang
- College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, P. R. China
| | - Zhidong Teng
- College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, P. R. China
| | - Tingting Zheng
- College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, P. R. China
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14
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Sahoo D, Samanta GP. Comparison between two tritrophic food chain models with multiple delays and anti-predation effect. INT J BIOMATH 2021. [DOI: 10.1142/s1793524521500108] [Citation(s) in RCA: 13] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
Exploring the predator–prey linkage in food chain system is the most familiar research work in population biology. Recently, some research experiments show that predator–prey interaction not only governed by direct hunting but also influenced by some indirect effect such as fear effect (felt by prey) that may change the physiological behavior of prey. Based upon this fact, we consider a tritrophic food chain model incorporating with anti-predation response (fear effect) and multiple time delays for biomass conversion from prey to middle predator and middle to top predator. We analyze the resulting delay differential equations and explore how the anti-predation response level affects the population dynamics. We also investigate the effect of delay parameters, for which the model system switches its stability through Hopf-bifurcation. We compare all of our results between two different food chain models consisting of two different functional responses. Some numerical simulations are performed to validate the effectiveness of the derived theoretical results.
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Affiliation(s)
- Debgopal Sahoo
- Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah-711103, India
| | - G. P. Samanta
- Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah-711103, India
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15
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Tuerxun N, Abdurahman X, Teng Z. Global dynamics and optimal harvesting in a stochastic two-predators one-prey system with distributed delays and Lévy noise. JOURNAL OF BIOLOGICAL DYNAMICS 2020; 14:32-56. [PMID: 31894726 DOI: 10.1080/17513758.2019.1707888] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/12/2019] [Accepted: 12/17/2019] [Indexed: 06/10/2023]
Abstract
In this paper, we first investigate a stochastic two-predators one-prey model with Lévy noise and distributed delays. The global dynamical behaviour is discussed. The criteria on the existence of global positive solutions, stochastic boundedness, extinction and global asymptotic stability in the mean with probability one are established. And then, the harvesting for each species is introduced to the model. The optimal harvesting strategy and the maximum of expectation of sustainable yield (MESY, for short) are further established.
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Affiliation(s)
- Nafeisha Tuerxun
- College of Mathematics and System Sciences, Xinjiang University, Urumqi, Xinjiang, People's Republic of China
| | - Xamxinur Abdurahman
- College of Mathematics and System Sciences, Xinjiang University, Urumqi, Xinjiang, People's Republic of China
| | - Zhidong Teng
- College of Mathematics and System Sciences, Xinjiang University, Urumqi, Xinjiang, People's Republic of China
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16
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Zafar ZUA, Ali N, Shah Z, Zaman G, Roy P, Deebani W. Hopf bifurcation and global dynamics of time delayed Dengue model. COMPUTER METHODS AND PROGRAMS IN BIOMEDICINE 2020; 195:105530. [PMID: 32531718 DOI: 10.1016/j.cmpb.2020.105530] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/10/2020] [Revised: 05/02/2020] [Accepted: 05/04/2020] [Indexed: 06/11/2023]
Abstract
BACKGROUND AND OBJECTIVE Dengue viral infections are a standout amongst the supreme critical mosquito-borne illnesses nowadays. They create problems like dengue fever (DF), dengue stun disorder (DSS) and dengue hemorrhagic fever (DHF). Lately, the frequency of DHF has expanded considerably. Dengue may be caused by one of serotypes DEN-1 to DEN-4. For the most part, septicity with one serotype presents upcoming defensive resistance against that specific serotype yet not against different serotypes. When anyone is infected for a second time with different serotypes, a serious ailment will occur. The proposed model focused on the dynamic interaction between susceptible cells and free virus cells. The ailment free steady states of the specimen are determined. The steadiness of the steady states has been examined by using Laplace transform. METHODS We introduce an appropriate numerical technique based on an Adams Bash-forth Moulton method for non-integer order delay differential equations. The numerical simulations validate the accuracy and efficacy of the numerical method. RESULTS In this paper, we study a non-integer order model with temporal delay to elaborate the dynamics of Dengue internal transmission dynamics. The temporal delay is presented in the susceptible cell and free virus cell. Centered on non-integer Laplace transform, some environs on firmness and Hopf bifurcation are derived for the model. Beside these global stability analysis is also done. Lastly, the imitative theoretical results are justified by few numerical simulations. CONCLUSION The study spectacles that the non-integer order with temporal-delay can successfully enhance the dynamics and rejuvenate the steadiness terms of non-integer order septicity prototypes. Both the ailment free equilibrium (AFE) node and ailment persistent equilibrium (APE) node are steady for the given system. We deduce a recipe that regulates the critical value at threshold.
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Affiliation(s)
- Zain Ul Abadin Zafar
- Faculty of Information Technology, University of Central Punjab, Lahore, Pakistan
| | - Nigar Ali
- Department of Mathematics, University of Malakand, Chakdara, Dir(Lower),Khyber Pakhtunkhwa, Pakistan
| | - Zahir Shah
- Center of Excellence in Theoretical and Computational Science (TaCS-CoE), SCL 802 Fixed Point Laboratory, Science Laboratory Building, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand.
| | - Gul Zaman
- Department of Mathematics, University of Malakand, Chakdara, Dir(Lower),Khyber Pakhtunkhwa, Pakistan
| | - Prosun Roy
- Department of Mechanical Engineering, University of Wisconsin, Milwaukee, WI, USA
| | - Wejdan Deebani
- Department of Mathematics, College of Science & Arts, King Abdulaziz University, P.O. Box 344, Rabigh 21911, Saudi Arabia
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17
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Jiang X, Zhang R, She Z. Dynamics of a diffusive predator–prey system with ratio-dependent functional response and time delay. INT J BIOMATH 2020. [DOI: 10.1142/s1793524520500369] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
In this paper, we investigate the qualitative behaviors of a predator–prey system with ratio-dependent function. The system accommodates the diffusion effect to model the migration of individuals and the time delay induced by reproduction. We start with some basic properties of the system. Then the sufficient condition independent of time delay and diffusion effect for global asymptotical stability of the boundary equilibrium is obtained by using the comparison principle. Afterwards, based on the LaSalle’s invariance principle and Lyapunov functional, we investigate the global attractiveness of the positive equilibrium, arriving at its global asymptotical stability. Further, Hopf bifurcation induced by time delay around the positive equilibrium is explored. Finally, numerical examples are listed to verify the corresponding analytical results.
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Affiliation(s)
- Xin Jiang
- School of Mathematical Sciences, Beihang University, Beijing 100191, P. R. China
| | - Ran Zhang
- School of Mathematics, Harbin Institute of Technology, Harbin 150001, P. R. China
| | - Zhikun She
- School of Mathematical Sciences, Beihang University, Beijing 100191, P. R. China
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18
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MAJI BANAMALI, TIWARI PANKAJKUMAR, SAMANTA SUDIP, PAL SAMARES, BONA FRANCESCA. EFFECT OF TIME DELAY IN A CANNIBALISTIC STAGE-STRUCTURED PREDATOR–PREY MODEL WITH HARVESTING OF AN ADULT PREDATOR: THE CASE OF LIONFISH. J BIOL SYST 2019. [DOI: 10.1142/s0218339019500189] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
The progressive and increasing invasion of an opportunistic predator, the lionfish (Pterois volitans) has become a major threat for the delicate coral-reef ecosystem. The herbivore fish populations, in particular of Parrotfish, are taking the consequences of the lionfish invasion and then their control function on macro-algae growth is threatened. In this paper, we developed and analyzed a stage-structured mathematical model including P. volitans (lionfish), a cannibalistic predator, and a Parrotfish, its potential prey. As control upon the over predation, a rational harvest term has been considered. Further, to make the system more realistic, a delay in the growth rate of juvenile P. volitans population has been incorporated. We performed a global sensitivity analysis to identify important parameters of the system having significant correlations with the fishes. We observed that the system generates transcritical bifurcation, which takes the P. volitans-free equilibrium to the coexistence equilibrium on increasing the values of predation rate of adult P. volitans on Parrotfish. Further increase in the values of the predation rate of adult P. volitans on Parrotfish drives the system into Hopf bifurcation, which induces oscillation around the coexistence equilibrium. Moreover, the conversion efficiency due to cannibalism also has the property to alter the stability behavior of the system through Hopf bifurcation. The effect of time delay on the dynamics of the system is extensively studied and it is observed that the system develops chaotic dynamics through period-doubling oscillations for large values of time delay. However, if the system is already oscillatory, then the large values of time delay causes extinction of P. volitans from the system. To illustrate the occurrence of chaotic dynamics in the system, we drew the Poincaré map and also computed the Lyapunov exponents.
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Affiliation(s)
- BANAMALI MAJI
- Department of Mathematics, Nayagram Pandit Raghunath Murmu Government College, Nayagram, Baligeria, Jhargram – 721125, India
| | | | - SUDIP SAMANTA
- Department of Mathematics, Bankura University, Bankura – 722155, West Bengal, India
| | - SAMARES PAL
- Department of Mathematics, University of Kalyani, Kalyani – 741235, India
| | - FRANCESCA BONA
- DBIOS, University of Turin, via Accademia Albertina 13, 10123 Turin, Italy
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Ghosh B, Pal D, Legović T, Kar TK. Harvesting induced stability and instability in a tri-trophic food chain. Math Biosci 2018; 304:89-99. [PMID: 30077688 DOI: 10.1016/j.mbs.2018.08.003] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/22/2018] [Revised: 07/31/2018] [Accepted: 08/01/2018] [Indexed: 11/24/2022]
Abstract
Non-equilibrium dynamics in the form of oscillations or chaos is often found to be a natural phenomenon in complex ecological systems. In this paper, we first analyze a tri-trophic food chain, which is an extension of the Rosenzweig-MacArthur di-trophic food chain. We then explore the impact of harvesting individual trophic levels to answer the following questions : a) when a non-equilibrium dynamics persists, b) whether it can locally be stabilized to a steady state, c) when the system switches from a stable steady state to a non-equilibrium dynamics and d) whether the Maximum Sustainable Yield (MSY) always exists when the top predator is harvested. It is shown that searching for a general theory to unify the harvesting induced stability must take into account the number of trophic levels and the degree of species enrichment, the outcomes that cannot be obtained from the earlier reports on prey-predator models. We also identify the situation where harvesting induces instability switching: the non-equilibrium state enters into a stable steady-state and then, upon more intensive harvesting, the steady-state again loses its stability. One of the new and important results is also that the MSY may not exist for harvesting the top predator. In general, our results contribute to biological conservation theory, fishery and ecosystem biodiversity management.
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Affiliation(s)
- Bapan Ghosh
- Department of Mathematics, National Institute of Technology Meghalaya, Bijni Complex, Shillong, Meghalaya 793003, India.
| | - Debprasad Pal
- Department of Mathematics, Indian Institute of Engineering Science and Technology Shibpur, Botanic Garden, Howrah, West Bengal 711103, India.
| | - Tarzan Legović
- Division of Marine and Environmental Research, R. Bošković Institute, POB 180, Zagreb HR-1002, Croatia.
| | - T K Kar
- Department of Mathematics, Indian Institute of Engineering Science and Technology Shibpur, Botanic Garden, Howrah, West Bengal 711103, India.
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20
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Bifurcation of a delayed Gause predator-prey model with Michaelis-Menten type harvesting. J Theor Biol 2018; 438:116-132. [PMID: 29129548 DOI: 10.1016/j.jtbi.2017.11.007] [Citation(s) in RCA: 23] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/08/2017] [Revised: 10/28/2017] [Accepted: 11/08/2017] [Indexed: 11/20/2022]
Abstract
In this paper, a Gause predator-prey model with gestation delay and Michaelis-Menten type harvesting of prey is proposed and analyzed by considering Holling type III functional response. We first consider the local stability of the interior equilibrium by investigating the corresponding characteristic equation. In succession, we derive some sufficient conditions on the occurrence of the stability switches of the positive steady state by taking the gestation delay as a bifurcation parameter. It is shown that the delay can induce instability and small amplitude oscillations of population densities via Hopf bifurcations. Furthermore, the stability and direction of the Hopf bifurcations are determined by employing the center manifold argument. Finally, computer simulations are performed to illustrate our analytical findings, and the biological implications of our analytical findings are also discussed.
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21
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Influence of predator standby capacity, harvesting and noise on a two patchy aquatic delayed eco system with migration of prey. INFORMATICS IN MEDICINE UNLOCKED 2018. [DOI: 10.1016/j.imu.2018.06.001] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/17/2022] Open
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Abstract
In this paper, a differential-algebraic predator–prey system with time delay is investigated, where the time delay is regarded as a parameter. By analyzing the corresponding characteristic equations, the local stability of the positive equilibrium and the existence of Hopf bifurcation are demonstrated. Furthermore, the explicit formulae which determine the stability, direction and other properties of bifurcating periodic solutions are obtained by applying the normal form theory and the center manifold argument. At last, some numerical simulations are carried out to illustrate the feasibility of our main results.
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Affiliation(s)
- Wei Liu
- School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, P. R. China
- School of Mathematics and Computer Science, Xinyu University, Xinyu 338004, P. R. China
| | - Yaolin Jiang
- School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, P. R. China
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23
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Banerjee M, Takeuchi Y. Maturation delay for the predators can enhance stable coexistence for a class of prey–predator models. J Theor Biol 2017; 412:154-171. [DOI: 10.1016/j.jtbi.2016.10.016] [Citation(s) in RCA: 30] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/22/2016] [Revised: 10/17/2016] [Accepted: 10/24/2016] [Indexed: 11/24/2022]
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24
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Hindes J, Szwaykowska K, Schwartz IB. Hybrid dynamics in delay-coupled swarms with mothership networks. Phys Rev E 2016; 94:032306. [PMID: 27739837 DOI: 10.1103/physreve.94.032306] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/17/2016] [Indexed: 06/06/2023]
Abstract
Swarming behavior continues to be a subject of immense interest because of its centrality in many naturally occurring systems in physics and biology, as well as its importance in applications such as robotics. Here we examine the effects on swarm pattern formation from delayed communication and topological heterogeneity, and in particular, the inclusion of a relatively small number of highly connected nodes, or "motherships," in a swarm's communication network. We find generalized forms of basic patterns for networks with general degree distributions, and a variety of dynamic behaviors including parameter regions with multistability and hybrid motions in bimodal networks. The latter is an interesting example of how heterogeneous networks can have dynamics that is a mix of different states in homogeneous networks, where high- and low-degree nodes have distinct behavior simultaneously.
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Affiliation(s)
- Jason Hindes
- U.S. Naval Research Laboratory, Code 6792, Plasma Physics Division, Nonlinear Dynamical Systems Section, Washington, DC 20375, USA
| | - Klementyna Szwaykowska
- U.S. Naval Research Laboratory, Code 6792, Plasma Physics Division, Nonlinear Dynamical Systems Section, Washington, DC 20375, USA
| | - Ira B Schwartz
- U.S. Naval Research Laboratory, Code 6792, Plasma Physics Division, Nonlinear Dynamical Systems Section, Washington, DC 20375, USA
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25
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Banerjee M, Zhang L. Time delay can enhance spatio-temporal chaos in a prey–predator model. ECOLOGICAL COMPLEXITY 2016. [DOI: 10.1016/j.ecocom.2015.12.001] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/15/2022]
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26
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A Simple Predator-Prey Population Model with Rich Dynamics. APPLIED SCIENCES-BASEL 2016. [DOI: 10.3390/app6050151] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
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27
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Szwaykowska K, Schwartz IB, Mier-Y-Teran Romero L, Heckman CR, Mox D, Hsieh MA. Collective motion patterns of swarms with delay coupling: Theory and experiment. Phys Rev E 2016; 93:032307. [PMID: 27078366 DOI: 10.1103/physreve.93.032307] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/21/2015] [Indexed: 06/05/2023]
Abstract
The formation of coherent patterns in swarms of interacting self-propelled autonomous agents is a subject of great interest in a wide range of application areas, ranging from engineering and physics to biology. In this paper, we model and experimentally realize a mixed-reality large-scale swarm of delay-coupled agents. The coupling term is modeled as a delayed communication relay of position. Our analyses, assuming agents communicating over an Erdös-Renyi network, demonstrate the existence of stable coherent patterns that can be achieved only with delay coupling and that are robust to decreasing network connectivity and heterogeneity in agent dynamics. We also show how the bifurcation structure for emergence of different patterns changes with heterogeneity in agent acceleration capabilities and limited connectivity in the network as a function of coupling strength and delay. Our results are verified through simulation as well as preliminary experimental results of delay-induced pattern formation in a mixed-reality swarm.
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Affiliation(s)
- Klementyna Szwaykowska
- U.S. Naval Research Laboratory, Code 6792, Plasma Physics Division, Nonlinear Dynamical Systems Section, Washington, DC, USA
| | - Ira B Schwartz
- U.S. Naval Research Laboratory, Code 6792, Plasma Physics Division, Nonlinear Dynamical Systems Section, Washington, DC, USA
| | - Luis Mier-Y-Teran Romero
- Johns Hopkins University, Bloomberg School of Public Health, Department of Epidemiology, Baltimore, Maryland, USA
| | | | - Dan Mox
- Drexel University, Mechanical Engineering & Mechanics Department, Scalable Autonomous Systems Laboratory, Philadelphia, Pennsylvania, USA
| | - M Ani Hsieh
- Drexel University, Mechanical Engineering & Mechanics Department, Scalable Autonomous Systems Laboratory, Philadelphia, Pennsylvania, USA
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28
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Zhang T, Zang H. Delay-induced Turing instability in reaction-diffusion equations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:052908. [PMID: 25493859 DOI: 10.1103/physreve.90.052908] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/25/2014] [Indexed: 06/04/2023]
Abstract
Time delays have been commonly used in modeling biological systems and can significantly change the dynamics of these systems. Quite a few works have been focused on analyzing the effect of small delays on the pattern formation of biological systems. In this paper, we investigate the effect of any delay on the formation of Turing patterns of reaction-diffusion equations. First, for a delay system in a general form, we propose a technique calculating the critical value of the time delay, above which a Turing instability occurs. Then we apply the technique to a predator-prey model and study the pattern formation of the model due to the delay. For the model in question, we find that when the time delay is small it has a uniform steady state or irregular patterns, which are not of Turing type; however, in the presence of a large delay we find spiral patterns of Turing type. For such a model, we also find that the critical delay is a decreasing function of the ratio of carrying capacity to half saturation of the prey density.
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Affiliation(s)
- Tonghua Zhang
- Department of Mathematics, Swinburne University of Technology, Melbourne 3122, Victoria, Australia
| | - Hong Zang
- Hubei Key Lab of Intelligent Robot, Wuhan Institute of Technology, Wuhan 430073, China
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29
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Stage-structured ratio-dependent predator–prey models revisited: When should the maturation lag result in systems’ destabilization? ECOLOGICAL COMPLEXITY 2014. [DOI: 10.1016/j.ecocom.2014.02.001] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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30
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Fu D, Tan P, Kuznetsov A, Molkov YI. Chaos and robustness in a single family of genetic oscillatory networks. PLoS One 2014; 9:e90666. [PMID: 24667178 PMCID: PMC3965403 DOI: 10.1371/journal.pone.0090666] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/18/2013] [Accepted: 02/03/2014] [Indexed: 12/04/2022] Open
Abstract
Genetic oscillatory networks can be mathematically modeled with delay differential equations (DDEs). Interpreting genetic networks with DDEs gives a more intuitive understanding from a biological standpoint. However, it presents a problem mathematically, for DDEs are by construction infinitely-dimensional and thus cannot be analyzed using methods common for systems of ordinary differential equations (ODEs). In our study, we address this problem by developing a method for reducing infinitely-dimensional DDEs to two- and three-dimensional systems of ODEs. We find that the three-dimensional reductions provide qualitative improvements over the two-dimensional reductions. We find that the reducibility of a DDE corresponds to its robustness. For non-robust DDEs that exhibit high-dimensional dynamics, we calculate analytic dimension lines to predict the dependence of the DDEs’ correlation dimension on parameters. From these lines, we deduce that the correlation dimension of non-robust DDEs grows linearly with the delay. On the other hand, for robust DDEs, we find that the period of oscillation grows linearly with delay. We find that DDEs with exclusively negative feedback are robust, whereas DDEs with feedback that changes its sign are not robust. We find that non-saturable degradation damps oscillations and narrows the range of parameter values for which oscillations exist. Finally, we deduce that natural genetic oscillators with highly-regular periods likely have solely negative feedback.
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Affiliation(s)
- Daniel Fu
- Department of Mathematics, Park Tudor School, Indianapolis, Indiana, United States of America
- * E-mail:
| | - Patrick Tan
- Department of Mathematics, Carmel High School, Carmel, Indiana, United States of America
| | - Alexey Kuznetsov
- Department of Mathematical Sciences, Indiana University-Purdue University of Indianapolis, Indianapolis, Indiana, United States of America
| | - Yaroslav I. Molkov
- Department of Mathematical Sciences, Indiana University-Purdue University of Indianapolis, Indianapolis, Indiana, United States of America
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31
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Stabilizing Effect of Prey Refuge and Predator’s Interference on the Dynamics of Prey with Delayed Growth and Generalist Predator with Delayed Gestation. INTERNATIONAL JOURNAL OF ECOLOGY 2014. [DOI: 10.1155/2014/429086] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
In the present paper, I study a prey-predator model with multiple time delays where the predator population is regarded as generalist. For this regard, I consider a Holling-Tanner prey-predator system where a constant time delay is incorporated in the logistic growth of the prey to represent a delayed density dependent feedback mechanism and the second time delay is considered to account for the length of the gestation period of the predator. Predator’s interference in predator-prey relationship provides better descriptions of predator's feeding over a range of prey-predator abundances, so the predator's functional response here is considered to be Type II ratio-dependent. In accordance with previous studies, it is observed that delay destabilizes the system, in general, and stability loss occurs via Hopf bifurcation. There exist critical values of delay parameters below which the coexistence equilibrium is stable and above which it is unstable. Hopf bifurcation occurs when delay parameters cross their critical values. When delay parameters are large enough than their critical values, the system exhibits chaotic behavior and this abnormal behavior may be controlled by refuge. Numerical computation is also performed to validate different theoretical results. Lyapunov exponent, recurrence plot, and power spectral density confirm the chaotic dynamical behaviors.
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32
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Tian C, Zhang L. Delay-driven irregular spatiotemporal patterns in a plankton system. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:012713. [PMID: 23944497 DOI: 10.1103/physreve.88.012713] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/02/2013] [Revised: 06/06/2013] [Indexed: 05/23/2023]
Abstract
An inhomogeneous distribution of species density over physical space is a widely observed scenario in plankton systems. Understanding the mechanisms resulting in these spatial patterns is a central topic in plankton ecology. In this paper we explore the impact of time delay on spatiotemporal patterns in a prey-predator plankton system. We find that time delay can trigger the emergence of irregular spatial patterns via a Hopf bifurcation. Moreover, a phase transition from a regular spiral pattern to an irregular one was observed and the latter gradually replaced the former and persisted indefinitely. The characteristic length of the emergent spatial pattern is consistent with the scale of plankton patterns observed in field studies.
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Affiliation(s)
- Canrong Tian
- Department of Basic Sciences, Yancheng Institute of Technology, Yancheng 224003, China.
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33
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Bhattacharyya J, Pal S. Stage-structured cannibalism with delay in maturation and harvesting of an adult predator. J Biol Phys 2013; 39:37-65. [PMID: 23860833 PMCID: PMC3532665 DOI: 10.1007/s10867-012-9284-6] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/05/2012] [Accepted: 08/20/2012] [Indexed: 10/27/2022] Open
Abstract
A three-dimensional stage-structured predator-prey model is proposed and analyzed to study the effect of predation and cannibalism of the organisms at the highest trophic level with non-constant harvesting. Time lag in maturation of the predator is introduced in the system and conditions for local asymptotic stability of steady states are derived. The length of the delay preserving the stability is also estimated. Moreover, it is shown that the system undergoes a supercritical Hopf bifurcation when the maturation time lag crosses a certain critical value. Computer simulations have been carried out to illustrate various analytical results.
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Affiliation(s)
| | - Samares Pal
- Department of Mathematics, University of Kalyani, Kalyani, 741235 India
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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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35
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MENG XINYOU, HUO HAIFENG, ZHANG XIAOBING. THE EFFECTS OF HARVESTING AND TIME DELAY ON PREDATOR-PREY SYSTEMS WITH BEDDINGTON–DEANGELIS FUNCTIONAL RESPONSE. INT J BIOMATH 2012. [DOI: 10.1142/s1793524511001489] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
The combined effects of harvesting and time delay on predator-prey systems with Beddington–DeAngelis functional response are studied. The region of stability in model with harvesting of the predator, local stability of equilibria and the existence of Hopf bifurcation are obtained by analyzing the associated characteristic equation due to the two-parameter geometric criteria developed by Ma, Feng and Lu [Discrete Contin. Dyn. Syst. Ser B9 (2008) 397–413]. The global stability of the positive equilibrium is investigated by the comparison theorem. Furthermore, local stability of steady states and the existence of Hopf bifurcation for prey harvesting are also considered. Numerical simulations are given to illustrate our theoretical findings.
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Affiliation(s)
- XINYOU MENG
- School of Science, Lanzhou University of Technology, Lanzhou, 730050, P. R. China
| | - HAIFENG HUO
- School of Science, Lanzhou University of Technology, Lanzhou, 730050, P. R. China
| | - XIAOBING ZHANG
- School of Science, Lanzhou University of Technology, Lanzhou, 730050, P. R. China
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Gupta RP, Banerjee M, Chandra P. The dynamics of two-species allelopathic competition with optimal harvesting. JOURNAL OF BIOLOGICAL DYNAMICS 2012; 6:674-694. [PMID: 22873612 DOI: 10.1080/17513758.2012.677484] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/01/2023]
Abstract
This paper analyses a bionomic model of two competitive species in the presence of toxicity with different harvesting efforts. An interesting dynamics in the first quadrant is analysed and two saddle-node bifurcations are detected for different bifurcation parameters. It is noted that under certain parametric restrictions, the model has a unique positive equilibrium point that is globally asymptotically stable whenever it is locally stable. It is also noted that the model can have zero, one or two feasible equilibria appearing through saddle-node bifurcations. The non-existence of a limit cycle in the interior of the first quadrant is also discussed using the Poincare-Dulac criteria. The saddle-node bifurcations are studied using Sotomayor's theorem. Numerical simulations are carried out to validate the analytical findings. The conditions for the existence of bionomic equilibria are discussed and an optimal harvesting policy is derived using Pontryagin's maximum principle.
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Affiliation(s)
- R P Gupta
- Department of Mathematics and Statistics, IIT Kanpur, Kanpur, 208016, India
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37
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LIU CHAO, ZHANG QINGLING, ZHANG XUE. DYNAMIC ANALYSIS IN A HARVESTED DIFFERENTIAL-ALGEBRAIC PREY–PREDATOR MODEL. J MECH MED BIOL 2011. [DOI: 10.1142/s0219519409002833] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
Nowadays, the biological resource in the prey–predator ecosystem is commercially harvested and sold with the aim of achieving economic interest. Furthermore, the harvest effort is usually influenced by the variation of economic interest of harvesting. In this paper, a differential–algebraic model is proposed, which is utilized to investigate the dynamical behavior of the prey–predator ecosystem due to the variation of economic interest of harvesting. By discussing the local stability of the proposed model around the interior equilibrium, the instability mechanism of harvested prey–predator ecosystem is studied. With the purpose of stabilizing the proposed model around the interior equilibrium and maintaining the economic interest of harvesting at an ideal level, a feedback controller is designed. Finally, numerical simulations are carried out to demonstrate consistency with the theoretical analysis.
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Affiliation(s)
- CHAO LIU
- Department of Information and Computational Science, Northeastern University of Qinhuangdao, Qinhuangdao 066004, P.R. China
| | - QINGLING ZHANG
- Institute of Systems Science, Key Laboratory of Integrated Automation of Process Industry, Ministry of Education, Northeastern University, Shenyang Liaoning 110004, China
| | - XUE ZHANG
- Institute of Systems Science, Key Laboratory of Integrated Automation of Process Industry, Ministry of Education, Northeastern University, Shenyang Liaoning 110004, China
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GAZI NURULHUDA, DAS KALYAN. CONTROL OF PARAMETERS OF A DELAYED DIFFUSIVE AUTOTROPH HERBIVORE SYSTEM. J BIOL SYST 2011. [DOI: 10.1142/s0218339010003305] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
This paper deals with the eco-biological dynamics of a delayed diffusive autotroph-herbivore population ecosystem with nutrient recycling. The real situation is represented by a set of two-dimensional nonlinear ordinary differential equations involving autotroph-herbivore biomass. Plant populations undergo critical changes with different amplitude in plant ecology. We propose a description of plant communities as interesting systems which resembles to the behavior of real media. The delay and diffusion parameter have a great role to shape the dynamical features of the system. We have studied the growth of an autotroph and herbivore population depending on the limiting nutrient which is partially recycled through bacterial decomposition. We have analyzed the asymptotic stability and switching to instability with bifurcation of the model system with delay and diffusion where diffusion controls directly the delay parameters of our system. All the analytical results are interpreted ecologically and compared with the simulated computer results.
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Affiliation(s)
- NURUL HUDA GAZI
- St. Xavier's College, 30, Park Street, Kolkata-700 016, India
| | - KALYAN DAS
- Department of Mathematics, B. P. Poddar Institute of Management and Technology, 137, V. I. P. Road, Kolkata 700 052, India
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39
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Delay Stochastic Simulation of Biological Systems: A Purely Delayed Approach. LECTURE NOTES IN COMPUTER SCIENCE 2011. [DOI: 10.1007/978-3-642-19748-2_4] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/13/2022]
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Harvesting as a disease control measure in an eco-epidemiological system – A theoretical study. Math Biosci 2009; 217:134-44. [DOI: 10.1016/j.mbs.2008.11.002] [Citation(s) in RCA: 50] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/22/2007] [Revised: 10/27/2008] [Accepted: 11/07/2008] [Indexed: 11/18/2022]
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