1
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Franco D, Perán J, Segura J. New insights into the combined effect of dispersal and local dynamics in a two-patch population model. J Theor Biol 2024; 595:111942. [PMID: 39299679 DOI: 10.1016/j.jtbi.2024.111942] [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: 04/20/2024] [Revised: 07/23/2024] [Accepted: 09/07/2024] [Indexed: 09/22/2024]
Abstract
Understanding the effect of dispersal on fragmented populations has drawn the attention of ecologists and managers in recent years, and great efforts have been made to understand the impact of dispersal on the total population size. All previous numerical and theoretical findings determined that the possible response scenarios of the overall population size to increasing dispersal are monotonic or hump-shaped, which has become a common assumption in ecology. Against this, we show in this paper that many other response scenarios are possible by using a simple two-patch discrete-time model. This fact evidences the interplay of local dynamics and dispersal and has significant consequences from a management perspective that will be discussed.
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Affiliation(s)
- Daniel Franco
- Department of Applied Mathematics, E.T.S.I. Industriales, Universidad Nacional de Educación a Distancia (UNED), c/ Juan del Rosal 12, Madrid, 28040, Spain.
| | - Juan Perán
- Department of Applied Mathematics, E.T.S.I. Industriales, Universidad Nacional de Educación a Distancia (UNED), c/ Juan del Rosal 12, Madrid, 28040, Spain.
| | - Juan Segura
- Department of Finance and Management Control, EADA Business School, c/Aragó, 204, Barcelona, 08011, Spain.
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2
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Nguyen TD, Wu Y, Tang T, Veprauskas A, Zhou Y, Rouhani BD, Shuai Z. Impact of resource distributions on the competition of species in stream environment. J Math Biol 2023; 87:62. [PMID: 37736867 DOI: 10.1007/s00285-023-01978-6] [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: 01/01/2023] [Revised: 07/27/2023] [Accepted: 07/31/2023] [Indexed: 09/23/2023]
Abstract
Our earlier work in Nguyen et al. (Maximizing metapopulation growth rate and biomass in stream networks. arXiv preprint arXiv:2306.05555 , 2023) shows that concentrating resources on the upstream end tends to maximize the total biomass in a metapopulation model for a stream species. In this paper, we continue our research direction by further considering a Lotka-Volterra competition patch model for two stream species. We show that the species whose resource allocations maximize the total biomass has the competitive advantage.
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Affiliation(s)
- Tung D Nguyen
- Department of Mathematics, Texas A &M University, College Station, TX, 77843, USA
| | - Yixiang Wu
- Department of Mathematical Sciences, Middle Tennessee State University, Murfreesboro, TN, 37132, USA.
| | - Tingting Tang
- Department of Mathematics and Statistics, San Diego State University, San Diego, CA, 92182, USA
| | - Amy Veprauskas
- Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA, 70501, USA
| | - Ying Zhou
- Department of Mathematics, Lafayette College, Easton, PA, 18042, USA
| | - Behzad Djafari Rouhani
- Department of Mathematical Sciences, University of Texas at El Paso, El Paso, TX, 79968, USA
| | - Zhisheng Shuai
- Department of Mathematics, University of Central Florida, Orlando, FL, 32816, USA
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3
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Grumbach C, Reurik FN, Segura J, Franco D, Hilker FM. The effect of dispersal on asymptotic total population size in discrete- and continuous-time two-patch models. J Math Biol 2023; 87:60. [PMID: 37733146 PMCID: PMC10514157 DOI: 10.1007/s00285-023-01984-8] [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: 01/20/2023] [Revised: 06/04/2023] [Accepted: 08/09/2023] [Indexed: 09/22/2023]
Abstract
Many populations occupy spatially fragmented landscapes. How dispersal affects the asymptotic total population size is a key question for conservation management and the design of ecological corridors. Here, we provide a comprehensive overview of two-patch models with symmetric dispersal and two standard density-dependent population growth functions, one in discrete and one in continuous time. A complete analysis of the discrete-time model reveals four response scenarios of the asymptotic total population size to increasing dispersal rate: (1) monotonically beneficial, (2) unimodally beneficial, (3) beneficial turning detrimental, and (4) monotonically detrimental. The same response scenarios exist for the continuous-time model, and we show that the parameter conditions are analogous between the discrete- and continuous-time setting. A detailed biological interpretation offers insight into the mechanisms underlying the response scenarios that thus improve our general understanding how potential conservation efforts affect population size.
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Affiliation(s)
- Carolin Grumbach
- Institute of Mathematics and Institute of Environmental Systems Research, Osnabrück University, Barbarastraße 12, 49076 Osnabrück, Germany
| | - Femke N. Reurik
- Institute of Mathematics and Institute of Environmental Systems Research, Osnabrück University, Barbarastraße 12, 49076 Osnabrück, Germany
| | - Juan Segura
- Department of Finance & Management Control, EADA Business School, c/ Aragó 204, 08011 Barcelona, Spain
| | - Daniel Franco
- Department of Applied Mathematics, UNED, c/ Juan del Rosal 12, 28040 Madrid, Spain
| | - Frank M. Hilker
- Institute of Mathematics and Institute of Environmental Systems Research, Osnabrück University, Barbarastraße 12, 49076 Osnabrück, Germany
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4
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Lou Y, Salako RB, Song P. Human mobility and disease prevalence. J Math Biol 2023; 87:20. [PMID: 37392280 DOI: 10.1007/s00285-023-01953-1] [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: 02/27/2023] [Revised: 05/19/2023] [Accepted: 06/12/2023] [Indexed: 07/03/2023]
Abstract
We examine the effect of human mobility on disease prevalence by studying the dependence of the total infected population at endemic equilibria with respect to population diffusion rates of a diffusive epidemic model. For small diffusion rates, our results indicate that the total infected population size is strictly decreasing with respect to the ratio of the diffusion rate of the infected population over that of the susceptible population. Moreover, when the disease local reproductive function is spatially heterogeneous, we found that: (i) for large diffusion rate of the infected population, the total infected population size is strictly maximized at large diffusion rate of the susceptible population when the recovery rate is spatially homogeneous, while it is strictly maximized at intermediate diffusion rate of the susceptible population when the difference of the transmission and recovery rates are spatially homogeneous; (ii) for large diffusion rate of the susceptible population, the total infected population size is strictly maximized at intermediate diffusion rate of the infected population when the recovery rate is spatially homogeneous, while it is strictly minimized at large diffusion rate of the infected population when the difference of the transmission and recovery rates is spatially homogeneous. Numerical simulations are provided to complement the theoretical results. Our studies may provide some insight into the impact of human mobility on disease outbreaks and the severity of epidemics.
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Affiliation(s)
- Yuan Lou
- School of Mathematical Sciences, CMA-Shanghai and MOE-LSC, Shanghai Jiao Tong University, Shanghai, 200240, China
| | - Rachidi B Salako
- Department of Mathematical Sciences, University of Nevada at Las Vegas, Las Vegas, NV, 89154, USA.
| | - Pengfei Song
- School of Mathematics and Statistics, Xi'an Jiaotong University, Shaanxi, 710049, China
- Laboratory for Industrial and Applied Mathematics, Department of Mathematics and Statistics, York University, Toronto, ON, M3J1P3, Canada
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5
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Elbetch B, Moussaoui A. Nonlinear diffusion in multi-patch logistic model. J Math Biol 2023; 87:1. [PMID: 37280354 DOI: 10.1007/s00285-023-01936-2] [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: 09/07/2022] [Revised: 02/10/2023] [Accepted: 05/18/2023] [Indexed: 06/08/2023]
Abstract
We examine a multi-patch model of a population connected by nonlinear asymmetrical migration, where the population grows logistically on each patch. Utilizing the theory of cooperative differential systems, we prove the global stability of the model. In cases of perfect mixing, where migration rates approach infinity, the total population follows a logistic law with a carrying capacity that is distinct from the sum of carrying capacities and is influenced by migration terms. Furthermore, we establish conditions under which fragmentation and nonlinear asymmetrical migration can lead to a total equilibrium population that is either greater or smaller than the sum of carrying capacities. Finally, for the two-patch model, we classify the model parameter space to determine if nonlinear dispersal is beneficial or detrimental to the sum of two carrying capacities.
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Affiliation(s)
- Bilel Elbetch
- Department of Mathematics, University Dr. Moulay Tahar of Saida, Saida, Algeria
| | - Ali Moussaoui
- Laboratoire d'Analyse Non linéaire et Mathématiques Appliquées, Department of Mathematics, University of Tlemcen, Chetouane, Algeria.
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6
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Valega-Mackenzie W, Bintz J, Lenhart S. Resource allocation in a PDE ecosystem model. J Math Biol 2023; 86:96. [PMID: 37217639 DOI: 10.1007/s00285-023-01932-6] [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: 06/01/2022] [Revised: 05/01/2023] [Accepted: 05/03/2023] [Indexed: 05/24/2023]
Abstract
The effects of habitat heterogeneity on a diffusing population are investigated here. We formulate a reaction-diffusion system of partial differential equations to analyze the effect of resource allocation in an ecosystem with resource having its own dynamics in space and time. We show a priori estimates to prove the existence of state solutions given a control. We formulate an optimal control problem of our ecosystem model such that the abundance of a single species is maximized while minimizing the cost of inflow resource allocation. In addition, we show the existence and uniqueness of the optimal control as well as the optimal control characterization. We also establish the existence of an optimal intermediate diffusion rate. Moreover, we illustrate several numerical simulations with Dirichlet and Neumann boundary conditions with the space domain in 1D and 2D.
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Affiliation(s)
| | - Jason Bintz
- School of Arts and Sciences, Johnson University, Knoxville, 37998, TN, USA
| | - Suzanne Lenhart
- Department of Mathematics, University of Tennessee Knoxville, Knoxville, 37996, TN, USA
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7
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Ruiz-Herrera A. The role of the spatial topology in trophic metacommunities: Species with reduced mobility and total population size. J Theor Biol 2023; 566:111479. [PMID: 37075827 DOI: 10.1016/j.jtbi.2023.111479] [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: 10/26/2022] [Revised: 02/10/2023] [Accepted: 03/24/2023] [Indexed: 04/21/2023]
Abstract
A central question in ecology is understanding the influence of the spatial topology on the dynamics of a metacommunity. This is not an easy task, as most fragmented ecosystems have trophic interactions involving many species and patches. Recent attempts to solve this challenge have introduced certain simplifying assumptions or focused on a limited set of examples. These simplifications make the models mathematically tractable but keep away from real-world problems. In this paper, we provide a novel methodology to describe the influence of the spatial topology on the total population size of the species when the dispersal rates are small. The main conclusion is that the influence of the spatial topology is the result of the influence of each path in isolation. Here, a path refers to a pairwise connection between two patches. Our framework can be readily used with any metacommunity, and therefore represents a unification of biological insights. We also discuss several applications regarding the construction of ecological corridors.
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8
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Lou Y, Sun B. Stage duration distributions and intraspecific competition: a review of continuous stage-structured models. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2022; 19:7543-7569. [PMID: 35801435 DOI: 10.3934/mbe.2022355] [Citation(s) in RCA: 4] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/15/2023]
Abstract
Stage structured models, by grouping individuals with similar demographic characteristics together, have proven useful in describing population dynamics. This manuscript starts from reviewing two widely used modeling frameworks that are in the form of integral equations and age-structured partial differential equations. Both modeling frameworks can be reduced to the same differential equation structures with/without time delays by applying Dirac and gamma distributions for the stage durations. Each framework has its advantages and inherent limitations. The net reproduction number and initial growth rate can be easily defined from the integral equation. However, it becomes challenging to integrate the density-dependent regulations on the stage distribution and survival probabilities in an integral equation, which may be suitably incorporated into partial differential equations. Further recent modeling studies, in particular those by Stephen A. Gourley and collaborators, are reviewed under the conditions of the stage duration distribution and survival probability being regulated by population density.
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Affiliation(s)
- Yijun Lou
- Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong SAR, China
| | - Bei Sun
- Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong SAR, China
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9
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Gao D, Lou Y. Total biomass of a single population in two-patch environments. Theor Popul Biol 2022; 146:1-14. [DOI: 10.1016/j.tpb.2022.05.003] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/08/2021] [Revised: 04/17/2022] [Accepted: 05/13/2022] [Indexed: 11/16/2022]
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10
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Gao D, Lou Y. Impact of State-Dependent Dispersal on Disease Prevalence. JOURNAL OF NONLINEAR SCIENCE 2021; 31:73. [PMID: 34248287 PMCID: PMC8254459 DOI: 10.1007/s00332-021-09731-3] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 10/07/2020] [Accepted: 06/23/2021] [Indexed: 06/13/2023]
Abstract
Based on a susceptible-infected-susceptible patch model, we study the influence of dispersal on the disease prevalence of an individual patch and all patches at the endemic equilibrium. Specifically, we estimate the disease prevalence of each patch and obtain a weak order-preserving result that correlated the patch reproduction number with the patch disease prevalence. Then we assume that dispersal rates of the susceptible and infected populations are proportional and derive the overall disease prevalence, or equivalently, the total infection size at no dispersal or infinite dispersal as well as the right derivative of the total infection size at no dispersal. Furthermore, for the two-patch submodel, two complete classifications of the model parameter space are given: one addressing when dispersal leads to higher or lower overall disease prevalence than no dispersal, and the other concerning how the overall disease prevalence varies with dispersal rate. Numerical simulations are performed to further investigate the effect of movement on disease prevalence.
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Affiliation(s)
- Daozhou Gao
- Department of Mathematics, Shanghai Normal University, Shanghai, 200234 China
| | - Yuan Lou
- School of Mathematical Sciences, Shanghai Jiaotong University, Shanghai, 200240 China
- Department of Mathematics, Ohio State University, Columbus, OH 43210 USA
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11
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Wu H, Wang Y. Dynamics of Competitive Systems with Diffusion Between Source-Sink Patches. Bull Math Biol 2021; 83:49. [PMID: 33765224 DOI: 10.1007/s11538-021-00885-5] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/28/2020] [Accepted: 03/08/2021] [Indexed: 11/27/2022]
Abstract
This paper considers two-species competitive systems with one-species' diffusion between patches. Each species can persist alone in the corresponding patch (a source), while the mobile species cannot survive in the other (a sink). Using the method of monotone dynamical systems, we give a rigorous analysis on persistence of the system, prove local/global stability of the equilibria and show new types of bi-stability. These results demonstrate that diffusion could lead to results reversing those without diffusion, which extend the principle of competitive exclusion: Diffusion could lead to persistence of the mobile competitor in the sink, make it reach total abundance larger than if non-diffusing and even exclude the opponent. The total abundance is shown to be a distorted function (surface) of diffusion rates, which extends both previous theory and experimental observations. A novel strategy of diffusion is deduced in which the mobile competitor could drive the opponent into extinction, and then approach the maximal abundance. Initial population density and diffusive asymmetry play a role in the competition. Our work has potential applications in biodiversity conservation and economic competition.
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Affiliation(s)
- Hong Wu
- School of Mathematics, Sun Yat-sen University, Guangzhou, 510275, People's Republic of China
| | - Yuanshi Wang
- School of Mathematics, Sun Yat-sen University, Guangzhou, 510275, People's Republic of China.
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12
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Xiao S, Wang Y, Wang S. Effects of Prey's Diffusion on Predator-Prey Systems with Two Patches. Bull Math Biol 2021; 83:45. [PMID: 33745081 DOI: 10.1007/s11538-021-00884-6] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/17/2020] [Accepted: 03/08/2021] [Indexed: 11/24/2022]
Abstract
This paper considers predator-prey systems in which the prey can move between source and sink patches. First, we give a complete analysis on global dynamics of the model. Then, we show that when diffusion from the source to sink is not large, the species would coexist at a steady state; when the diffusion is large, the predator goes to extinction, while the prey persists in both patches at a steady state; when the diffusion is extremely large, both species go to extinction. It is derived that diffusion in the system could lead to results reversing those without diffusion. That is, diffusion could change species' coexistence if non-diffusing, to extinction of the predator, and even to extinction of both species. Furthermore, we show that intermediate diffusion to the sink could make the prey reach total abundance higher than if non-diffusing, larger or smaller diffusion rates are not favorable. The total abundance, as a function of diffusion rates, can be both hump-shaped and bowl-shaped, which extends previous theory. A novel finding of this work is that there exist diffusion scenarios which could drive the predator into extinction and make the prey reach the maximal abundance. Diffusion from the sink to source and asymmetry in diffusion could also lead to results reversing those without diffusion. Meanwhile, diffusion always leads to reduction of the predator's density. The results are biologically important in protection of endangered species.
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Affiliation(s)
- Siheng Xiao
- School of Mathematics, Sun Yat-sen University, 510275, Guangzhou, People's Republic of China
| | - Yuanshi Wang
- School of Mathematics, Sun Yat-sen University, 510275, Guangzhou, People's Republic of China.
| | - Shikun Wang
- School of Mathematics, Sun Yat-sen University, 510275, Guangzhou, People's Republic of China.,Department of Biostatistics, The University of Texas MD Anderson Cancer Center, Houston, TX, 77030, USA
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13
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Carrying Capacity of Spatially Distributed Metapopulations. Trends Ecol Evol 2021; 36:164-173. [DOI: 10.1016/j.tree.2020.10.007] [Citation(s) in RCA: 9] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/18/2020] [Revised: 10/03/2020] [Accepted: 10/08/2020] [Indexed: 12/28/2022]
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14
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Huang R, Wang Y, Wu H. Population abundance in predator–prey systems with predator’s dispersal between two patches. Theor Popul Biol 2020; 135:1-8. [DOI: 10.1016/j.tpb.2020.06.002] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/30/2019] [Revised: 06/22/2020] [Accepted: 06/29/2020] [Indexed: 11/17/2022]
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15
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Zhang B, DeAngelis DL, Ni WM, Wang Y, Zhai L, Kula A, Xu S, Van Dyken JD. Effect of Stressors on the Carrying Capacity of Spatially Distributed Metapopulations. Am Nat 2020; 196:E46-E60. [DOI: 10.1086/709293] [Citation(s) in RCA: 14] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/04/2022]
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16
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On the effects of carrying capacity and intrinsic growth rate on single and multiple species in spatially heterogeneous environments. J Math Biol 2020; 81:403-433. [PMID: 32621114 DOI: 10.1007/s00285-020-01507-9] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/14/2019] [Revised: 05/04/2020] [Indexed: 10/23/2022]
Abstract
We first consider a diffusive logistic model of a single species in a heterogeneous environment, with two parameters, r(x) for intrinsic growth rate and K(x) for carrying capacity. When r(x) and K(x) are proportional, i.e., [Formula: see text], it is proved by Lou (J Differ Equ 223(2):400-426, 2006) that a population diffusing at any rate will reach a higher total equilibrium biomass than the population in an environment in which the same total resources are distributed homogeneously. This paper studies another case when r(x) is a constant, i.e., independent of K(x). In such case, a striking result is that for any dispersal rate, the logistic equation with spatially heterogeneous resources will always support a total population strictly smaller than the total carrying capacity at equilibrium, which is just opposite to the case [Formula: see text]. These two cases of single species models also lead to two different forms of Lotka-Volterra competition-diffusion systems. We then examine the consequences of the aforementioned difference on the two forms of competition systems. We find that the outcome of the competition in terms of the dispersal rates and spatial distributions of resources for the two forms of competition systems are again quite different. Our results indicate that in heterogeneous environments, the correlation between r(x) and K(x) has more profound impacts in population ecology than we had previously expected, at least from a mathematical point of view.
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17
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Wang Y, Wu H, He Y, Wang Z, Hu K. Population abundance of two-patch competitive systems with asymmetric dispersal. J Math Biol 2020; 81:315-341. [PMID: 32572557 DOI: 10.1007/s00285-020-01511-z] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/30/2019] [Revised: 05/22/2020] [Indexed: 11/24/2022]
Abstract
This paper considers two-species competitive systems with two patches, in which one of the species can move between the patches. One patch is a source where each species can persist alone, but the other is a sink where the mobile species cannot survive. Based on rigorous analysis on the model, we show global stability of equilibria and bi-stability in the first octant Int[Formula: see text]. Then total population abundance of each species is explicitly expressed as a function of dispersal rates, and the function of the mobile species displays a distorted surface, which extends previous theory. A novel prediction of this work is that appropriate dispersal could make each competitor approach total population abundance larger than if non-dispersing, while the dispersal could reverse the competitive results in the absence of dispersal and promote coexistence of competitors. It is also shown that intermediate dispersal is favorable, large or small one is not good, while extremely large or small dispersal will result in extinction of species. These results are important in ecological conservation and management.
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Affiliation(s)
- Yuanshi Wang
- School of Mathematics, Sun Yat-sen University, Guangzhou, 510275, People's Republic of China.
| | - Hong Wu
- School of Mathematics, Sun Yat-sen University, Guangzhou, 510275, People's Republic of China
| | - Yiyang He
- School of Mathematics, Sun Yat-sen University, Guangzhou, 510275, People's Republic of China
| | - Zhihui Wang
- School of Mathematics, Sun Yat-sen University, Guangzhou, 510275, People's Republic of China
| | - Kun Hu
- School of Mathematics, Sun Yat-sen University, Guangzhou, 510275, People's Republic of China
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18
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Ruiz-Herrera A, Torres PJ. Optimal Network Architectures for Spatially Structured Populations with Heterogeneous Diffusion. Am Nat 2020; 196:29-44. [PMID: 32552100 DOI: 10.1086/708806] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/03/2022]
Abstract
The motivation of this article is to derive new management guidelines to maximize the overall population size using popular management and conservation strategies, such as protected marine areas and ecological corridors. These guidelines are based on the identification of the network architectures for which the total population size is maximized. Describing the biological roles of the typical network variables in the fate of the population is a classic problem with many practical applications. This article suggests that the optimal network architecture relies heavily on the degree of mobility of the population. The recommended network architecture for populations with reduced mobility (in the absence of cost of dispersal and landscapes made up of many sources) is a graph with a patch that has routes toward any other patch with a lower growth rate. However, for highly mobile populations there are many possible network architectures for which the total population size is maximized (e.g., any cyclic graph). We have paid special attention to species with symmetric movement in heterogeneous landscapes. A striking result is that the network architecture does not have any influence on the total population size for highly mobile populations when any pair of different patches can be connected by a sequence of paths.
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19
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Abstract
The carrying capacity of the environment for a population is one of the key concepts in ecology and it is incorporated in the growth term of reaction-diffusion equations describing populations in space. Analysis of reaction-diffusion models of populations in heterogeneous space have shown that, when the maximum growth rate and carrying capacity in a logistic growth function vary in space, conditions exist for which the total population size at equilibrium (i) exceeds the total population that which would occur in the absence of diffusion and (ii) exceeds that which would occur if the system were homogeneous and the total carrying capacity, computed as the integral over the local carrying capacities, was the same in the heterogeneous and homogeneous cases. We review here work over the past few years that has explained these apparently counter-intuitive results in terms of the way input of energy or another limiting resource (e.g., a nutrient) varies across the system. We report on both mathematical analysis and laboratory experiments confirming that total population size in a heterogeneous system with diffusion can exceed that in the system without diffusion. We further report, however, that when the resource of the population in question is explicitly modeled as a coupled variable, as in a reaction-diffusion chemostat model rather than a model with logistic growth, the total population in the heterogeneous system with diffusion cannot exceed the total population size in the corresponding homogeneous system in which the total carrying capacities are the same.
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20
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The Effect of Movement Behavior on Population Density in Patchy Landscapes. Bull Math Biol 2019; 82:1. [DOI: 10.1007/s11538-019-00680-3] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/20/2019] [Accepted: 12/02/2019] [Indexed: 10/25/2022]
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21
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Wu H, Wang Y, Li Y, DeAngelis DL. Dispersal asymmetry in a two-patch system with source-sink populations. Theor Popul Biol 2019; 131:54-65. [PMID: 31778710 DOI: 10.1016/j.tpb.2019.11.004] [Citation(s) in RCA: 14] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/25/2019] [Revised: 10/18/2019] [Accepted: 11/15/2019] [Indexed: 11/26/2022]
Abstract
This paper analyzes source-sink systems with asymmetric dispersal between two patches. Complete analysis on the models demonstrates a mechanism by which the dispersal asymmetry can lead to either an increased total size of the species population in two patches, a decreased total size with persistence in the patches, or even extinction in both patches. For a large growth rate of the species in the source and a fixed dispersal intensity, (i) if the asymmetry is small, the population would persist in both patches and reach a density higher than that without dispersal, in which the population approaches its maximal density at an appropriate asymmetry; (ii) if the asymmetry is intermediate, the population persists in both patches but reaches a density less than that without dispersal; (iii) if the asymmetry is large, the population goes to extinction in both patches; (iv) asymmetric dispersal is more favorable than symmetric dispersal under certain conditions. For a fixed asymmetry, similar phenomena occur when the dispersal intensity varies, while a thorough analysis is given for the low growth rate of the species in the source. Implications for populations in heterogeneous landscapes are discussed, and numerical simulations confirm and extend our results.
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Affiliation(s)
- Hong Wu
- School of Mathematics, Sun Yat-sen University, Guangzhou 510275, PR China
| | - Yuanshi Wang
- School of Mathematics, Sun Yat-sen University, Guangzhou 510275, PR China.
| | - Yufeng Li
- School of Mathematics, Sun Yat-sen University, Guangzhou 510275, PR China
| | - Donald L DeAngelis
- U.S. Geological Survey, Wetland and Aquatic Research Center, Gainesville, FL 32653, USA
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22
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Wang Y. Pollination-mutualisms in a two-patch system with dispersal. J Theor Biol 2019; 476:51-61. [DOI: 10.1016/j.jtbi.2019.06.004] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/11/2019] [Revised: 06/01/2019] [Accepted: 06/04/2019] [Indexed: 11/17/2022]
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23
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Tan C, Wang Y, Wu H. Population abundance of a two-patch chemostat system with asymmetric diffusion. J Theor Biol 2019; 474:1-13. [PMID: 31054917 DOI: 10.1016/j.jtbi.2019.04.026] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/15/2019] [Revised: 04/25/2019] [Accepted: 04/30/2019] [Indexed: 10/26/2022]
Abstract
This paper considers a two-patch chemostat system with asymmetric diffusion, which characterizes laboratory experiments and includes exploitable nutrients. Using dynamical system theory, we demonstrate global stability of the one-patch model, and show uniform persistence of the two-patch system, which leads to existence of a stable positive equilibrium. Analysis on the equilibrium demonstrates mechanisms by which varying the asymmetric diffusion can make the total population abundance in heterogeneous environments larger than that without diffusion, even larger than that in the corresponding homogeneous environments with or without diffusion. The mechanisms are shown to be effective even in source-sink populations. A novel finding of this work is that the asymmetry combined with high diffusion intensity can reverse the predictions of symmetric diffusion in previous studies, while intermediate asymmetry is shown to be favorable but extremely large or extremely small asymmetry is unfavorable. Our results are consistent with experimental observations and provide new insights. Numerical simulations confirm and extend the results.
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Affiliation(s)
- Chengguan Tan
- School of Mathematics, Sun Yat-sen University, Guangzhou 510275, PR China
| | - Yuanshi Wang
- School of Mathematics, Sun Yat-sen University, Guangzhou 510275, PR China
| | - Hong Wu
- School of Mathematics, Sun Yat-sen University, Guangzhou 510275, PR China.
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24
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Wang Y. Asymptotic State of a Two-Patch System with Infinite Diffusion. Bull Math Biol 2019; 81:1665-1686. [DOI: 10.1007/s11538-019-00582-4] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/15/2018] [Accepted: 02/10/2019] [Indexed: 11/30/2022]
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25
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Wang Y, DeAngelis DL. Energetic constraints and the paradox of a diffusing population in a heterogeneous environment. Theor Popul Biol 2019; 125:30-37. [DOI: 10.1016/j.tpb.2018.11.003] [Citation(s) in RCA: 17] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/02/2017] [Revised: 10/18/2018] [Accepted: 11/27/2018] [Indexed: 11/25/2022]
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26
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Dynamics of a consumer-resource reaction-diffusion model : Homogeneous versus heterogeneous environments. J Math Biol 2019; 78:1605-1636. [PMID: 30603993 DOI: 10.1007/s00285-018-1321-z] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/29/2018] [Revised: 11/30/2018] [Indexed: 10/27/2022]
Abstract
We study the dynamics of a consumer-resource reaction-diffusion model, proposed recently by Zhang et al. (Ecol Lett 20(9):1118-1128, 2017), in both homogeneous and heterogeneous environments. For homogeneous environments we establish the global stability of constant steady states. For heterogeneous environments we study the existence and stability of positive steady states and the persistence of time-dependent solutions. Our results illustrate that for heterogeneous environments there are some parameter regions in which the resources are only partially limited in space, a unique feature which does not occur in homogeneous environments. Such difference between homogeneous and heterogeneous environments seems to be closely connected with a recent finding by Zhang et al. (2017), which says that in consumer-resource models, homogeneously distributed resources could support higher population abundance than heterogeneously distributed resources. This is opposite to the prediction by Lou (J Differ Equ 223(2):400-426, 2006. https://doi.org/10.1016/j.jde.2005.05.010 ) for logistic-type models. For both small and high yield rates, we also show that when a consumer exists in a region with a heterogeneously distributed input of exploitable renewed limiting resources, the total population abundance at equilibrium can reach a greater abundance when it diffuses than when it does not. In contrast, such phenomenon may fail for intermediate yield rates.
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27
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Carrying capacity of a spatially-structured population: Disentangling the effects of dispersal, growth parameters, habitat heterogeneity and habitat clustering. J Theor Biol 2018; 460:115-124. [PMID: 30253138 DOI: 10.1016/j.jtbi.2018.09.015] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/23/2018] [Revised: 09/07/2018] [Accepted: 09/16/2018] [Indexed: 11/22/2022]
Abstract
Carrying capacity, K, is a fundamental quantity in theoretical and applied ecology. When populations are distributed over space, carrying capacity becomes a complicated function of local, global and nearby environments, dispersal rate, and the relationship between population growth parameters, e.g., r and K. Expressions for the total carrying capacity, Ktotal, in an n-patch model that explicitly disentangle all of these factors are currently lacking. Therefore, here we derive Ktotal for a linear spatial array of n habitat patches with logistic growth and strong or weak random dispersal of individuals between adjacent patches. With strong dispersal, Ktotal depends on the mean r and K over all patches (〈r〉 and 〈K〉), the among-patch variance in K, and the linear regression coefficient of r on K, βr,K. Strong dispersal increases Ktotal only if βr, K > 〈r〉/〈K〉, which requires a positive convex or negative concave association between r and K, and decreases Ktotal if βr, K < 〈r〉/〈K〉. Alternatively, weak dispersal increases Ktotal only if the within-patch covariance of r and K, cov(r, K) is greater than the spatial covariance between r and K, cov(r, Km), defined as the average covariance between r in a focal patch and K in neighboring patches. Unlike the strong dispersal limit, this condition depends not only on the magnitude of environmental heterogeneity, but explicitly on the spatial distribution of heterogeneity (i.e., habitat clustering). This work clarifies how the interaction between dispersal, habitat heterogeneity, and population growth parameters shape carrying capacity in spatial populations, with implications for species management, conservation and evolution.
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28
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Ruiz-Herrera A, Torres PJ. Effects of diffusion on total biomass in simple metacommunities. J Theor Biol 2018; 447:12-24. [PMID: 29550452 DOI: 10.1016/j.jtbi.2018.03.018] [Citation(s) in RCA: 16] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/21/2017] [Revised: 03/01/2018] [Accepted: 03/13/2018] [Indexed: 11/16/2022]
Abstract
This paper analyzes the effects of diffusion on the overall population size of the different species of a metacommunity. Depending on precise thresholds, we determine whether increasing the dispersal rate of a species has a positive or negative effect on population abundance. These thresholds depend on the interaction type of the species and the quality of the patches. The motivation for researching this issue is that spatial structure is a source of new biological insights with management interest. For instance, in a metacommunity of two competitors, the movement of a competitor could lead to a decrease of the overall population size of both species. On the other hand, we discuss when some classic results of metapopulation theory are preserved in metacommunities. Our results complement some recent experimental work by Zhang and collaborators.
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Affiliation(s)
| | - Pedro J Torres
- Departamento de Matemática Aplicada, Universidad de Gradana, Spain.
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29
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Zhang B, Kula A, Mack KML, Zhai L, Ryce AL, Ni WM, DeAngelis DL, Van Dyken JD. Carrying capacity in a heterogeneous environment with habitat connectivity. Ecol Lett 2017; 20:1118-1128. [PMID: 28712141 DOI: 10.1111/ele.12807] [Citation(s) in RCA: 49] [Impact Index Per Article: 7.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/07/2017] [Revised: 03/25/2017] [Accepted: 06/03/2017] [Indexed: 11/28/2022]
Abstract
A large body of theory predicts that populations diffusing in heterogeneous environments reach higher total size than if non-diffusing, and, paradoxically, higher size than in a corresponding homogeneous environment. However, this theory and its assumptions have not been rigorously tested. Here, we extended previous theory to include exploitable resources, proving qualitatively novel results, which we tested experimentally using spatially diffusing laboratory populations of yeast. Consistent with previous theory, we predicted and experimentally observed that spatial diffusion increased total equilibrium population abundance in heterogeneous environments, with the effect size depending on the relationship between r and K. Refuting previous theory, however, we discovered that homogeneously distributed resources support higher total carrying capacity than heterogeneously distributed resources, even with species diffusion. Our results provide rigorous experimental tests of new and old theory, demonstrating how the traditional notion of carrying capacity is ambiguous for populations diffusing in spatially heterogeneous environments.
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Affiliation(s)
- Bo Zhang
- Department of Biology, University of Miami, Coral Gables, FL, USA
| | - Alex Kula
- Department of Biology, University of Miami, Coral Gables, FL, USA
| | - Keenan M L Mack
- Department of Biology, Illinois College, Jacksonville, IL, USA
| | - Lu Zhai
- Department of Biology, University of Miami, Coral Gables, FL, USA.,Department of Mathematics, University of Miami, Coral Gables, FL, USA
| | - Arrix L Ryce
- Department of Biology, University of Miami, Coral Gables, FL, USA
| | - Wei-Ming Ni
- School of Mathematics, University of Minnesota, Minneapolis, MN, USA.,Center for Partial Differential Equations, East China Normal University, Putuo Qu, Shanghai Shi, China
| | - Donald L DeAngelis
- Wetland and Aquatic Research Center, U.S. Geological Survey, Gainesville, FL, USA
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30
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DeAngelis DL, Ni WM, Zhang B. Dispersal and spatial heterogeneity: single species. J Math Biol 2015; 72:239-54. [PMID: 25862553 DOI: 10.1007/s00285-015-0879-y] [Citation(s) in RCA: 35] [Impact Index Per Article: 3.9] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/20/2014] [Revised: 03/15/2015] [Indexed: 11/30/2022]
Abstract
A recent result for a reaction-diffusion equation is that a population diffusing at any rate in an environment in which resources vary spatially will reach a higher total equilibrium biomass than the population in an environment in which the same total resources are distributed homogeneously. This has so far been proven by Lou for the case in which the reaction term has only one parameter, m(x), varying with spatial location x, which serves as both the intrinsic growth rate coefficient and carrying capacity of the population. However, this striking result seems rather limited when applies to real populations. In order to make the model more relevant for ecologists, we consider a logistic reaction term, with two parameters, r (x) for intrinsic growth rate, and K(x) for carrying capacity. When r (x) and K(x) are proportional, the logistic equation takes a particularly simple form, and the earlier result still holds. In this paper we have established the result for the more general case of a positive correlation between r (x) and K(x) when dispersal rate is small. We review natural and laboratory systems to which these results are relevant and discuss the implications of the results to population theory and conservation ecology.
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Affiliation(s)
- Donald L DeAngelis
- Department of Biology, University of Miami, Coral Gables, FL, 33124, USA.
| | - Wei-Ming Ni
- Center for Partial Differential Equations, East China Normal University, Minhang, Shanghai, 200241, People's Republic of China. .,School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA.
| | - Bo Zhang
- Department of Biology, University of Miami, Coral Gables, FL, 33124, USA.
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