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Hughes EH, Moyers-Gonzalez M, Murray R, Wilson PL. Partial differential equation models for invasive species spread in the presence of spatial heterogeneity. PLoS One 2024; 19:e0300968. [PMID: 38564572 PMCID: PMC10986974 DOI: 10.1371/journal.pone.0300968] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/30/2023] [Accepted: 03/06/2024] [Indexed: 04/04/2024] Open
Abstract
Models of invasive species spread often assume that landscapes are spatially homogeneous; thus simplifying analysis but potentially reducing accuracy. We extend a recently developed partial differential equation model for invasive conifer spread to account for spatial heterogeneity in parameter values and introduce a method to obtain key outputs (e.g. spread rates) from computational simulations. Simulations produce patterns of spatial spread which appear qualitatively similar to observed patterns in grassland ecosystems invaded by exotic conifers, validating our spatially explicit strategy. We find that incorporating spatial variation in different parameters does not significantly affect the evolution of invasions (which are characterised by a long quiescent period followed by rapid evolution towards to a constant rate of invasion) but that distributional assumptions can have a significant impact on the spread rate of invasions. Our work demonstrates that spatial variation in site-suitability or other parameters can have a significant impact on invasions and must be considered when designing models of invasive species spread.
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Affiliation(s)
- Elliott H. Hughes
- School of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand
- Mathematical Institute, University of Oxford, Oxford, United Kingdom
| | - Miguel Moyers-Gonzalez
- School of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand
| | - Rua Murray
- School of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand
| | - Phillip L. Wilson
- School of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand
- Te Pūnaha Matatini Centre of Research Excellence, University of Auckland, Auckland, New Zealand
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2
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Li B, Otto G. Spread and persistence for integro-difference equations with shifting habitat and strong Allee effect. J Math Biol 2024; 88:35. [PMID: 38427042 DOI: 10.1007/s00285-024-02048-1] [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: 09/29/2023] [Accepted: 12/30/2023] [Indexed: 03/02/2024]
Abstract
We study an integro-difference equation model that describes the spatial dynamics of a species with a strong Allee effect in a shifting habitat. We examine the case of a shifting semi-infinite bad habitat connected to a semi-infinite good habitat. In this case we rigorously establish species persistence (non-persistence) if the habitat shift speed is less (greater) than the asymptotic spreading speed of the species in the good habitat. We also examine the case of a finite shifting patch of hospitable habitat, and find that the habitat shift speed must be less than the asymptotic spreading speed associated with the habitat and there is a critical patch size for species persistence. Spreading speeds and traveling waves are established to address species persistence. Our numerical simulations demonstrate the theoretical results and show the dependence of the critical patch size on the shift speed.
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Affiliation(s)
- Bingtuan Li
- Department of Mathematics, University of Louisville, Louisville, KY, 40292, USA.
| | - Garrett Otto
- Department of Mathematics, SUNY Cortland, Cortland, NY, 13045, USA
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4
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Wave speed and critical patch size for integro-difference equations with a strong Allee effect. J Math Biol 2022; 85:59. [PMID: 36273068 DOI: 10.1007/s00285-022-01814-3] [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: 07/09/2021] [Revised: 04/18/2022] [Accepted: 04/19/2022] [Indexed: 10/24/2022]
Abstract
Simplified conditions are given for the existence and positivity of wave speed for an integro-difference equation with a strong Allee effect and an unbounded habitat. The results are used to obtain the existence of a critical patch size for an equation with a bounded habitat. It is shown that if the wave speed is positive there exists a critical patch size such that for a habitat size above the critical patch size solutions can persist in space, and if the wave speed is negative solutions always approach zero. An analytical integral formula is developed to determine the critical patch size when the Laplace dispersal kernel is used, and this formula shows existence of multiple equilibrium solutions. Numerical simulations are provided to demonstrate connections among the wave speed, critical patch size, and Allee threshold.
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Shen W, Shen Z, Xue S, Zhou D. Population dynamics under climate change: persistence criterion and effects of fluctuations. J Math Biol 2022; 84:30. [PMID: 35274161 DOI: 10.1007/s00285-022-01728-0] [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: 01/18/2021] [Revised: 09/11/2021] [Accepted: 02/07/2022] [Indexed: 11/29/2022]
Abstract
The present paper is devoted to the investigation of population dynamics under climate change. The evolution of species is modelled by a reaction-diffusion equation in a spatio-temporally heterogeneous environment described by a climate envelope that shifts with a time-dependent speed function. For a general almost-periodic speed function, we establish the persistence criterion in terms of the sign of the approximate top Lyapunov exponent and, in the case of persistence, prove the existence of a unique forced wave solution that dominates the population profile of species in the long run. In the setting for studying the effects of fluctuations in the shifting speed or location of the climate envelope, we show by means of matched asymptotic expansions and numerical simulations that the approximate top Lyapunov exponent is a decreasing function with respect to the amplitude of fluctuations, yielding that fluctuations in the shifting speed or location have negative impacts on the persistence of species, and moreover, the larger the fluctuation is, the more adverse the effect is on the species. In addition, we assert that large fluctuations can always drive a species to extinction. Our numerical results also show that a persistent species under climate change is invulnerable to mild fluctuations, and becomes vulnerable when fluctuations are so large that the species is endangered. Finally, we show that fluctuations of amplitude less than or equal to the speed difference between the shifting speed and the critical speed are too weak to endanger a persistent species.
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Affiliation(s)
- Wenxian Shen
- Department of Mathematics and Statistics, Auburn University, Auburn, AL, 36849, USA
| | - Zhongwei Shen
- Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada
| | - Shuwen Xue
- Department of Mathematics and Statistics, Auburn University, Auburn, AL, 36849, USA.
| | - Dun Zhou
- School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing, 210094, Jiangsu, People's Republic of China
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Zhou Y. Range Shifts Under Constant-Speed and Accelerated Climate Warming. Bull Math Biol 2021; 84:1. [PMID: 34787723 DOI: 10.1007/s11538-021-00963-8] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/31/2020] [Accepted: 10/25/2021] [Indexed: 11/29/2022]
Abstract
Many species are experiencing range shifts as the climate warms. Earlier models have considered moving-habitat scenarios where the shift speed of the moving habitat is either constant or fluctuates around a constant speed. This paper considers scenarios where the shifting speed may either be constant or accelerated. In addition to population persistence, the paper also analyzes the range-shift deficit, which is a lag in the spatial distribution of the population. Another highlight of the results is an analytic formula for a population persistence metric in a particular example, which adds to existing analytic formulas about this persistence metric in the literature.
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Affiliation(s)
- Ying Zhou
- Department of Mathematics, Lafayette College, Easton, USA.
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MacDonald JS, Bourgault Y, Lutscher F. Moving-habitat models: A numerical approach. Math Biosci 2021; 341:108711. [PMID: 34547364 DOI: 10.1016/j.mbs.2021.108711] [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: 05/03/2021] [Revised: 09/09/2021] [Accepted: 09/09/2021] [Indexed: 10/20/2022]
Abstract
As the global climate changes, biological populations have to adapt in place or move in space to stay within their preferred temperature regime. Empirical evidence suggests that shifting speeds of temperature isoclines are location and elevation dependent and may accelerate over time. We present a mathematical tool to study transient behaviour of population dynamics within such moving habitats to discern between populations at high and low risk of extinction. We introduce a system of reaction-diffusion equations to study the impact of varying shifting speeds on the persistence and distribution of a single species. Our model includes habitat dependent movement behaviour and habitat preference of individuals. These assumptions result in a jump in density across habitat types and generalize previous studies. We build and validate a numerical finite difference scheme to solve the resulting equations. Our numerical scheme uses a coordinate system where the location of the moving suitable habitat is fixed in space and a modification of a finite difference scheme to capture the jump in density. We explore a variety of shifting-speed scenarios and contribute insights into the mechanisms that support population persistence through time in shifting habitats. One common finding is that a strong bias for the suitable habitat helps the population persist at faster shifting speeds, yet sustains a smaller total population at slower shifting speeds.
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Affiliation(s)
- Jane S MacDonald
- University of Ottawa, Department of Mathematics and Statistics, 150 Louis-Pasteur Pvt, K1N 6N5, Ottawa, Canada.
| | - Yves Bourgault
- University of Ottawa, Department of Mathematics and Statistics, 150 Louis-Pasteur Pvt, K1N 6N5, Ottawa, Canada.
| | - Frithjof Lutscher
- University of Ottawa, Department of Mathematics and Statistics, 150 Louis-Pasteur Pvt, K1N 6N5, Ottawa, Canada; University of Ottawa, Department of Biology, Canada.
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9
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Discrete-time population dynamics of spatially distributed semelparous two-sex populations. J Math Biol 2021; 83:18. [PMID: 34322725 DOI: 10.1007/s00285-021-01649-4] [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: 08/15/2020] [Revised: 06/10/2021] [Accepted: 07/18/2021] [Indexed: 10/20/2022]
Abstract
Spatially distributed populations with two sexes may face the problem that males and females concentrate in different parts of the habitat and mating and reproduction does not happen sufficiently often for the population to persist. For simplicity, to explore the impact of sex-dependent dispersal on population survival, we consider a discrete-time model for a semelparous population where individuals reproduce only once in their life-time, during a very short reproduction season. The dispersal of females and males is modeled by Feller kernels and the mating by a homogeneous pair formation function. The spectral radius of a homogeneous operator is established as basic reproduction number of the population, [Formula: see text]. If [Formula: see text], the extinction state is locally stable, and if [Formula: see text] the population shows various degrees of persistence that depend on the irreducibility properties of the dispersal kernels. Special cases exhibit how sex-biased dispersal affects the persistence of the population.
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Miller TEX, Angert AL, Brown CD, Lee-Yaw JA, Lewis M, Lutscher F, Marculis NG, Melbourne BA, Shaw AK, Szűcs M, Tabares O, Usui T, Weiss-Lehman C, Williams JL. Eco-evolutionary dynamics of range expansion. Ecology 2020; 101:e03139. [PMID: 32697876 DOI: 10.1002/ecy.3139] [Citation(s) in RCA: 45] [Impact Index Per Article: 11.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Received: 01/15/2020] [Revised: 05/05/2020] [Accepted: 06/08/2020] [Indexed: 01/31/2023]
Abstract
Understanding the movement of species' ranges is a classic ecological problem that takes on urgency in this era of global change. Historically treated as a purely ecological process, range expansion is now understood to involve eco-evolutionary feedbacks due to spatial genetic structure that emerges as populations spread. We synthesize empirical and theoretical work on the eco-evolutionary dynamics of range expansion, with emphasis on bridging directional, deterministic processes that favor evolved increases in dispersal and demographic traits with stochastic processes that lead to the random fixation of alleles and traits. We develop a framework for understanding the joint influence of these processes in changing the mean and variance of expansion speed and its underlying traits. Our synthesis of recent laboratory experiments supports the consistent role of evolution in accelerating expansion speed on average, and highlights unexpected diversity in how evolution can influence variability in speed: results not well predicted by current theory. We discuss and evaluate support for three classes of modifiers of eco-evolutionary range dynamics (landscape context, trait genetics, and biotic interactions), identify emerging themes, and suggest new directions for future work in a field that stands to increase in relevance as populations move in response to global change.
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Affiliation(s)
- Tom E X Miller
- Program in Ecology and Evolutionary Biology, Department of BioSciences, Rice University, Houston, Texas, 77005, USA
| | - Amy L Angert
- Department of Botany, University of British Columbia, Vancouver, British Columbia, V6T 1Z2, Canada
| | - Carissa D Brown
- Department of Geography, Memorial University, 230 Elizabeth Avenue, St John's, Newfoundland and Labrador, A1B 3X9, Canada
| | - Julie A Lee-Yaw
- Department of Botany, University of British Columbia, Vancouver, British Columbia, V6T 1Z2, Canada.,Department of Biological Sciences, University of Lethbridge, 4401 University Drive, Lethbridge, Alberta, T1K 3M4, Canada
| | - Mark Lewis
- Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1, Canada.,Department of Biological Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1, Canada
| | - Frithjof Lutscher
- Department of Mathematics and Statistics, and Department of Biology, University of Ottawa, Ottawa, Ottawa, K1N 6N5, Canada
| | - Nathan G Marculis
- Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1, Canada.,Department of Environmental Science and Policy, University of California-Davis, Davis, California, 95616, USA
| | - Brett A Melbourne
- Department of Ecology and Evolutionary Biology, University of Colorado, Boulder, Colorado, 80309, USA
| | - Allison K Shaw
- Department of Ecology, Evolution, and Behavior, University of Minnesota, St. Paul, Minnesota, 55108, USA
| | - Marianna Szűcs
- Department of Entomology, Michigan State University, 288 Farm Lane, East Lansing, Michigan, 48824, USA
| | - Olivia Tabares
- Department of Geography and Biodiversity Research Centre, University of British Columbia, 1984 West Mall, Vancouver, British Columbia, V6T 1Z2, Canada
| | - Takuji Usui
- Department of Botany, University of British Columbia, Vancouver, British Columbia, V6T 1Z2, Canada
| | - Christopher Weiss-Lehman
- Department of Ecology, Evolution, and Behavior, University of Minnesota, St. Paul, Minnesota, 55108, USA
| | - Jennifer L Williams
- Department of Geography and Biodiversity Research Centre, University of British Columbia, 1984 West Mall, Vancouver, British Columbia, V6T 1Z2, Canada
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Marculis NG, Lewis MA. Inside Dynamics of Integrodifference Equations with Mutations. Bull Math Biol 2020; 82:7. [PMID: 31932985 DOI: 10.1007/s11538-019-00683-0] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/28/2019] [Accepted: 12/02/2019] [Indexed: 10/25/2022]
Abstract
The method of inside dynamics provides a theory that can track the dynamics of neutral gene fractions in spreading populations. However, the role of mutations has so far been absent in the study of the gene flow of neutral fractions via inside dynamics. Using integrodifference equations, we develop a neutral genetic mutation model by extending a previously established scalar inside dynamics model. To classify the mutation dynamics, we define a mutation class as the set of neutral fractions that can mutate into one another. We show that the spread of neutral genetic fractions is dependent on the leading edge of population as well as the structure of the mutation matrix. Specifically, we show that the neutral fractions that contribute to the spread of the population must belong to the same mutation class as the neutral fraction found in the leading edge of the population. We prove that the asymptotic proportion of individuals at the leading edge of the population spread is given by the dominant right eigenvector of the associated mutation matrix, independent of growth and dispersal parameters. In addition, we provide numerical simulations to demonstrate our mathematical results, to extend their generality and to develop new conjectures about our model.
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Affiliation(s)
- Nathan G Marculis
- Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada.
| | - Mark A Lewis
- Department of Mathematical and Statistical Sciences & Department of Biological Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada
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