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Hutchison C, Guichard F, Legagneux P, Gauthier G, Bêty J, Berteaux D, Fauteux D, Gravel D. Seasonal food webs with migrations: multi-season models reveal indirect species interactions in the Canadian Arctic tundra. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2020; 378:20190354. [PMID: 32862818 PMCID: PMC7481661 DOI: 10.1098/rsta.2019.0354] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/12/2023]
Abstract
Models incorporating seasonality are necessary to fully assess the impact of global warming on Arctic communities. Seasonal migrations are a key component of Arctic food webs that still elude current theories predicting a single community equilibrium. We develop a multi-season model of predator-prey dynamics using a hybrid dynamical systems framework applied to a simplified tundra food web (lemming-fox-goose-owl). Hybrid systems models can accommodate multiple equilibria, which is a basic requirement for modelling food webs whose topology changes with season. We demonstrate that our model can generate multi-annual cycling in lemming dynamics, solely from a combined effect of seasonality and state-dependent behaviour. We compare our multi-season model to a static model of the predator-prey community dynamics and study the interactions between species. Interestingly, including seasonality reveals indirect interactions between migrants and residents not captured by the static model. Further, we find that the direction and magnitude of interactions between two species are not necessarily accurate using only summer time-series. Our study demonstrates the need for the development of multi-season models and provides the tools to analyse them. Integrating seasonality in food web modelling is a vital step to improve predictions about the impacts of climate change on ecosystem functioning. This article is part of the theme issue 'The changing Arctic Ocean: consequences for biological communities, biogeochemical processes and ecosystem functioning'.
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Affiliation(s)
| | | | - Pierre Legagneux
- Département de Biologie et Centre d’Études Nordiques, Université Laval, Québéc City, Canada
- Centre d’Études Biologiques de Chizé, CNRS-la Rochelle Université, Villiers-en-Bois, France
| | - Gilles Gauthier
- Département de Biologie et Centre d’Études Nordiques, Université Laval, Québéc City, Canada
| | - Joël Bêty
- Département de Biologie et Centre d’Études nordiques, Université du Québec à Rimouski, Rimouski, Canada
| | - Dominique Berteaux
- Département de Biologie et Centre d’Études nordiques, Université du Québec à Rimouski, Rimouski, Canada
| | - Dominique Fauteux
- Département de Biologie et Centre d’Études Nordiques, Université Laval, Québéc City, Canada
- Canadian Museum of Nature, Ottawa, Canada
| | - Dominique Gravel
- Département de Biologie, Université de Sherbrooke, Sherbrooke, Canada
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Savoca S, Grifó G, Panarello G, Albano M, Giacobbe S, Capillo G, Spanó N, Consolo G. Modelling prey-predator interactions in Messina beachrock pools. Ecol Modell 2020. [DOI: 10.1016/j.ecolmodel.2020.109206] [Citation(s) in RCA: 15] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
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3
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Xu Y, Krause AL, Van Gorder RA. Generalist predator dynamics under kolmogorov versus non-Kolmogorov models. J Theor Biol 2020; 486:110060. [PMID: 31689420 DOI: 10.1016/j.jtbi.2019.110060] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/29/2019] [Revised: 08/14/2019] [Accepted: 10/30/2019] [Indexed: 10/25/2022]
Abstract
Ecosystems often contain multiple species across two or more trophic levels, with a variety of interactions possible. In this paper we study two classes of models for generalist predators that utilize more than one food source. These models fall into two categories: predator - two prey and predator - prey - subsidy models. For the former, we consider a generalist predator which utilizes two distinct prey species, modelled via a Kolmogorov system of equations with Type II response functions. For the latter, we consider a generalist predator which exploits both a prey population and an allochthonous resource which is provided as a subsidy to the system exogenously, again with Type II response functions. This latter class of model is no longer Kolmogorov in form, due to an exogenous forcing term modelling the input of the allochthonous resource into the system. We non-dimensionalize both models, so that their respective parameter spaces may be more easily compared, and study the dynamics possible from each type of model, which will then indicate - for specific parameter regimes - which generalist predator's preferences are more favorable to survival, including the prevalence of coexistence states. We also consider the various non-equilibrium dynamics emergent from such models, and show that the non-Kolmogorov predator - prey - subsidy model of 10 admits more regular dynamics (including steady states and one type of limit cycle), whereas the predator - two prey Kolmogorov model can feature multiple types of limit cycles, as well as multistability resulting in strong sensitivity to initial conditions (with stable limit cycles and steady states both coexisting for the same model parameters). Our results highlight several interesting differences and similarities between Kolmogorov and non-Kolmogorov models for generalist predators.
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Affiliation(s)
- Yifang Xu
- Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK
| | - Andrew L Krause
- Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK
| | - Robert A Van Gorder
- Department of Mathematics and Statistics, University of Otago, P.O. Box 56, Dunedin 9054, New Zealand.
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Narang A, Bhandary S, Kaur T, Gupta A, Banerjee T, Dutta PS. Long-range dispersal promotes species persistence in climate extremes. CHAOS (WOODBURY, N.Y.) 2019; 29:103136. [PMID: 31675831 DOI: 10.1063/1.5120105] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/16/2019] [Accepted: 10/07/2019] [Indexed: 06/10/2023]
Abstract
Anthropogenic global warming in this century can act as a leading factor for large scale species extinctions in the near future. Species, in order to survive, need to develop dispersal strategies depending upon their environmental niche. Based on empirical evidence only a few previous studies have addressed how dispersal can evolve with changing temperature. However, for the analytical tractability, there is a need to develop an explicit model to ask how the temperature-dependent dispersal alters ecological dynamics. We investigate the persistence of species in a spatial ecological model, where dispersal is considered as a function of temperature. Spatial persistence is of major concern and dispersal is reasonably an important factor for extinction risk in the context of promoting synchrony. Our study yields how the temperature influences species decision of dispersal, resulting in either short-range or long-range dispersal. We examine synchronous or asynchronous behavior of species under their thermal dependence of dispersal. Moreover, we also analyze the transients to study the collective behavior of species away from their final or asymptotic dynamics. One of the key findings is at the most unfavorable environmental conditions long-range dispersal works out as the driving force for the persistence of species.
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Affiliation(s)
- Arzoo Narang
- Department of Mathematics, Indian Institute of Technology Ropar, Rupnagar 140 001, Punjab, India
| | - Subhendu Bhandary
- Department of Mathematics, Indian Institute of Technology Ropar, Rupnagar 140 001, Punjab, India
| | - Taranjot Kaur
- Department of Mathematics, Indian Institute of Technology Ropar, Rupnagar 140 001, Punjab, India
| | - Anubhav Gupta
- Department of Evolutionary Biology and Environmental Studies, University of Zurich, 8057 Zurich, Switzerland
| | - Tanmoy Banerjee
- Chaos and Complex Systems Research Laboratory, Department of Physics, University of Burdwan, Burdwan 713 104, West Bengal, India
| | - Partha Sharathi Dutta
- Department of Mathematics, Indian Institute of Technology Ropar, Rupnagar 140 001, Punjab, India
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5
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Wang X, Wang H, Li MY. R 0 and sensitivity analysis of a predator-prey model with seasonality and maturation delay. Math Biosci 2019; 315:108225. [PMID: 31283915 DOI: 10.1016/j.mbs.2019.108225] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/20/2019] [Revised: 07/04/2019] [Accepted: 07/05/2019] [Indexed: 11/25/2022]
Abstract
Coexistence and seasonal fluctuations of predator and prey populations are common and well documented in ecology. Under what conditions can predators coexist with prey in a seasonally changing environment? What factors drive long-term population cycles of some predator and prey species? To answer these questions, we investigate an improved predator-prey model based on the Rosenzweig-MacArthur [1] model. Our model incorporates seasonality and a predator maturation delay, leading to a system of periodic differential equations with a time delay. We define the basic reproduction ratio R0 and show that it is a threshold parameter determining whether the predators can coexist with the prey. We show that if R0 < 1, then the prey population has seasonal variations and the predator population goes extinct. If R0 > 1, then the prey and the predators coexist and fluctuate seasonally. As an example, we study a Daphnia-algae system and explore possible mechanisms for seasonal population cycles. Our numerical simulations indicate that seasonal Daphnia-algae cycles are attributed to seasonality rather than Daphnia maturation delay or Daphnia-algae interaction. The Daphnia maturation delay, the amplitude of algae growth rate and the amplitude of the carrying capacity are found to affect the amplitude of cycles and average population levels. Our sensitivity analysis shows that R0 is most sensitive to Daphnia death rate.
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Affiliation(s)
- Xiunan Wang
- Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1, Canada.
| | - Hao Wang
- Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1, Canada.
| | - Michael Y Li
- Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1, Canada.
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Fussell EF, Krause AL, Van Gorder RA. Hybrid approach to modeling spatial dynamics of systems with generalist predators. J Theor Biol 2019; 462:26-47. [PMID: 30385311 DOI: 10.1016/j.jtbi.2018.10.054] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/14/2018] [Revised: 10/06/2018] [Accepted: 10/29/2018] [Indexed: 11/16/2022]
Abstract
We consider hybrid spatial modeling approaches for ecological systems with a generalist predator utilizing a prey and either a second prey or an allochthonous resource. While spatial dispersion of populations is often modeled via stepping-stone (discrete spatial patches) or continuum (one connected spatial domain) formulations, we shall be interested in hybrid approaches which we use to reduce the dimension of certain components of the spatial domain, obtaining either a continuum model of varying spatial dimensions, or a mixed stepping-stone-continuum model. This approach results in models consisting of partial differential equations for some of the species which are coupled via reactive boundary conditions to lower dimensional partial differential equations or ordinary differential equations for the other species. In order to demonstrate the use of this approach, we consider two case studies. In the first case study, we consider a one-predator two-prey interaction between beavers, wolves and white-tailed deer in Voyageurs National Park. In the second case study, we consider predator-prey-allochthonous resource interactions between bears, berries and salmon on Kodiak Island. For each case study, we compare the results from the hybrid modeling approach with corresponding stepping-stone and continuum model results, highlighting benefits and limitations of the method. In some cases, we find that the hybrid modeling approach allows for solutions which are easier to simulate (akin to stepping-stone models) while maintaining seemingly more realistic spatial dynamics (akin to full continuum models).
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Affiliation(s)
- Elizabeth F Fussell
- Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK
| | - Andrew L Krause
- Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK
| | - Robert A Van Gorder
- Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK.
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7
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Eide RM, Krause AL, Fadai NT, Van Gorder RA. Predator-prey-subsidy population dynamics on stepping-stone domains with dispersal delays. J Theor Biol 2018; 451:19-34. [PMID: 29723541 DOI: 10.1016/j.jtbi.2018.04.038] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/16/2017] [Revised: 04/18/2018] [Accepted: 04/20/2018] [Indexed: 10/17/2022]
Abstract
We examine the role of the travel time of a predator along a spatial network on predator-prey population interactions, where the predator is able to partially or fully sustain itself on a resource subsidy. The impact of access to food resources on the stability and behaviour of the predator-prey-subsidy system is investigated, with a primary focus on how incorporating travel time changes the dynamics. The population interactions are modelled by a system of delay differential equations, where travel time is incorporated as discrete delay in the network diffusion term in order to model time taken to migrate between spatial regions. The model is motivated by the Arctic ecosystem, where the Arctic fox consumes both hunted lemming and scavenged seal carcass. The fox travels out on sea ice, in addition to quadrennially migrating over substantial distances. We model the spatial predator-prey-subsidy dynamics through a "stepping-stone" approach. We find that a temporal delay alone does not push species into extinction, but rather may stabilize or destabilize coexistence equilibria. We are able to show that delay can stabilize quasi-periodic or chaotic dynamics, and conclude that the incorporation of dispersal delay has a regularizing effect on dynamics, suggesting that dispersal delay can be proposed as a solution to the paradox of enrichment.
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Affiliation(s)
- Ragna M Eide
- Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK
| | - Andrew L Krause
- Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK
| | - Nabil T Fadai
- Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK
| | - Robert A Van Gorder
- Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK.
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9
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Bassett A, Krause AL, Van Gorder RA. Continuous dispersal in a model of predator–prey-subsidy population dynamics. Ecol Modell 2017. [DOI: 10.1016/j.ecolmodel.2017.02.017] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
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10
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Shen L, Van Gorder RA. Predator-prey-subsidy population dynamics on stepping-stone domains. J Theor Biol 2017; 420:241-258. [PMID: 28322876 DOI: 10.1016/j.jtbi.2017.03.013] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/21/2016] [Revised: 02/19/2017] [Accepted: 03/10/2017] [Indexed: 12/01/2022]
Abstract
Predator-prey-subsidy dynamics on stepping-stone domains are examined using a variety of network configurations. Our problem is motivated by the interactions between arctic foxes (predator) and lemmings (prey) in the presence of seal carrion (subsidy) provided by polar bears. We use the n-Patch Model, which considers space explicitly as a "Stepping Stone" system. We consider the role that the carrying capacity, predator migration rate, input subsidy rate, predator mortality rate, and proportion of predators surviving migration play in the predator-prey-subsidy population dynamics. We find that for certain types of networks, added mobility will help predator populations, allowing them to survive or coexist when they would otherwise go extinct if confined to one location, while in other situations (such as when sparsely distributed nodes in the network have few resources available) the added mobility will hurt the predator population. We also find that a combination of favourable conditions for the prey and subsidy can lead to the formation of limit cycles (boom and bust dynamic) from stable equilibrium states. These modifications to the dynamics vary depending on the specific network structure employed, highlighting the fact that network structure can strongly influence the predator-prey-subsidy dynamics in stepping-stone domains.
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Affiliation(s)
- Lulan Shen
- Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
| | - Robert A Van Gorder
- Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK.
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