Integrodifference models for persistence in temporally varying river environments

Jacobsen, Jon; Jin, Yu; Lewis, Mark A.

doi:10.1007/s00285-014-0774-y

Cited by 24 publications

(32 citation statements)

References 44 publications

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“…(H4) This statement follows from the continuity of K, uniform boundedness of Fα and fr and Hypothesis 5, details can be found in Jacobsen et al (2015)[Section 5.3]. (H5) From assumption 4(ii)c, we have that for all u > 0, f r (0)u ≥ fr(u).…”

Section: Discussionmentioning

confidence: 89%

“…Indeed in these two papers the authors prove that the persistence of the population depends on the magnitude of the metric Λ (Jacobsen et al, 2015) defined as the asymptotic growth rate of the linearised operator, or equivalently on the magnitude of the metric R (Hardin et al, 1988) defined as the asymptotic L ∞ norm of the linearised operator. The former metric defined by Jacobsen et al (2015) has more biological meaning and is easier to compute numerically, which is why we chose to use it in this paper. Our paper thus provides a mathematical metric to measure persistence of the population facing climate change and shifted range distribution in a temporally variable environment.…”

Section: Discussionmentioning

confidence: 99%

“…We can thus apply the Theorem 4.2 from Hardin et al (1988) to prove Theorem 1 and use Theorem 5.3 by Hardin et al (1988) and Theorem 2 by Jacobsen et al (2015) to prove Theorem 2. For weak of completeness we include the three Theorems from Hardin et al (1988) and Jacobsen et al (2015) below.…”

Section: Discussionmentioning

confidence: 99%

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Climate Change and Integrodifference Equations in a Stochastic Environment

Bouhours

Lewis

2016

Bull Math Biol

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Climate change impacts population distributions, forcing some species to migrate poleward if they are to survive and keep up with the suitable habitat that is shifting with the temperature isoclines. Previous studies have analyzed whether populations have the capacity to keep up with shifting temperature isoclines, and have mathematically determined the combination of growth and dispersal that is needed to achieve this. However, the rate of isocline movement can be highly variable, with much uncertainty associated with yearly shifts. The same is true for population growth rates. Growth rates can be variable and uncertain, even within suitable habitats for growth. In this paper we reanalyze the question of population persistence in the context of the uncertainty and variability in isocline shifts and rates of growth. Specifically, we employ a stochastic integrodifference equation model on a patch of suitable habitat that shifts poleward at a random rate. We derive a metric describing the asymptotic growth rate of the linearized operator of the stochastic model. This metric yields a threshold criterion for population persistence. We demonstrate that the variability in the yearly shift and in the growth rate has a significant negative effect on the persistence in the sense that it decreases the threshold criterion for population persistence. Mathematically, we show how the persistence metric can be connected to the principal eigenvalue problem for a related integral operator, at least for the case where isocline shifting speed is deterministic. Analysis of dynamics for the case where the dispersal kernel is Gaussian leads to the existence of a critical shifting speed, above which the population will go extinct, and below which the population will persist. This leads to clear bounds on rate of environmental change if the population is to persist. Finally we illustrate our different results for butterfly population using numerical simulations and demonstrate how increased variances in isocline shifts and growth rates translate into decreased likelihoods of persistence.

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Section: Discussionmentioning

confidence: 89%

Section: Discussionmentioning

confidence: 99%

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Climate Change and Integrodifference Equations in a Stochastic Environment

Bouhours

Lewis

2016

Bull Math Biol

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“…The network topology (i.e., the topological structure of a network), together with other The population persistence in a spatial population model has been described by uniform persistence, the (in)stability of the trivial extinction solution, and the critical domain size (minimal length of the habitat such that a species can persist); see e.g., [18, 20, 27-29, 37, 40, 44]. For a single species population in one-dimensional rivers, the persistence theory was established in a homogeneous environment in [32,37,53,55], in temporally periodically varying environments in [20], in temporally randomly varying environments in [19], and in spatially heterogeneous environment in [33]. For a benthic-drift population consisting of individuals drifting in water and individuals staying on the benthos, the critical domain size was studied in a spatially homogeneous river in [44] and in a river with alternating good and bad channels in [34].…”

Section: Introductionmentioning

confidence: 99%

Population Dynamics in River Networks

2019

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Natural rivers connect to each other to form networks. The geometric structure of a river network can significantly influence spatial dynamics of populations in the system. We consider a process-oriented model to describe population dynamics in river networks of trees, establish the fundamental theories of the corresponding parabolic problems and elliptic problems, derive the persistence threshold by using the principal eigenvalue of the eigenvalue problem, and define the net reproductive rate to describe population persistence or extinction. By virtue of numerical simulations, we investigate the effects of hydrological, physical, and biological factors, especially the structure of the river network, on population persistence. physical and hydrological features in a river network, can greatly influence the spatial distribution of the flow profile (including the flow velocity, water depth, etc.) in branches of the network. Moreover, the population dispersal vectors may be constrained by the network configuration and the flow profile, and species life history traits may depend on varying habitat conditions in the network. As a result, population distribution and persistence in river systems can be significantly affected by the network topology or structure, see e.g., [8,11,12,[14][15][16]46]. Then there arise interesting questions such as whether a population can persist in the desert streams of the southwestern United State while the streams are experiencing substantial natural drying trends [11], or whether dendritic geometry enhances dynamic stability of ecological systems [15] etc. Furthermore, other related dynamics in the network, such as the dynamics of water-born infectious diseases like cholera may also be greatly affected by the river network geometry (see e.g., [6]).Branches in a river network have been modeled as point nodes in a network of habitats in individual based models [11,16] and matrix population models for stage-structured populations [14,39]. However this oversimplifies the spatial heterogeneity of river networks. In a real river ecosystem, organisms mainly live in the branches of the network and the connections between branches (e.g., the network nodes) are mainly for population transitions from one branch to another. To take into account this realistic situation, in recent works [48-50], integro-differential equations and reactiondiffusion-advection equations were used to model population dynamics in river networks where the network branches, instead of the network nodes, are the main habitat for organisms. Here the river networks are modeled under the framework of metric graphs (or metric networks). A metric graph is a graph G = (V, E) with a set V of vertices and a set E of edges, such that each edge e ∈ E is associated with either a closed bounded interval. Mathematic notion of metric graphs was first introduced in the context of wave propagation on thin graph-like domains [5,25], and they are also called quantum graphs.The theories of parabolic and elliptic equations as well as the corr...

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“…Examples of populations that have been modeled with integrodifference equations include house finches in the eastern North America, cane toads in Australia, blowflies in South Africa, and Platte thistle [65,50,40,13]. Integrodifference equations have also been used to understand the following: migration of trees, plant invasions, evolution of flowering strategies, persistence of populations in fragmented habitats and river environments, invasion of structured populations, Allee effects, movement of individuals in patchy landscapes, and habitat shifting as a result of climate change [8,38,56,64,44,67,25,43,71]. For a recent review of ecological applications see [31].…”

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confidence: 99%

Optimal control of integrodifference equations in a pest-pathogen system

Martinez¹,

Lenhart²,

White³

2015

DCDS-B

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We develop the theory of optimal control for a system of integrodifference equations modelling a pest-pathogen system. Integrodifference equations incorporate continuous space into a system of discrete time equations. We design an objective functional to minimize the damaged cost generated by an invasive species and the cost of controlling the population with a pathogen. Existence, characterization, and uniqueness results for the optimal control and corresponding states have been completed. We use a forwardbackward sweep numerical method to implement our optimization which produces spatio-temporal control strategies for the gypsy moth case study.2010 Mathematics Subject Classification. Primary: 49J21, 92D30; Secondary: 92D40.

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Integrodifference models for persistence in temporally varying river environments

Cited by 24 publications

References 44 publications

Climate Change and Integrodifference Equations in a Stochastic Environment

Climate Change and Integrodifference Equations in a Stochastic Environment

Population Dynamics in River Networks

Optimal control of integrodifference equations in a pest-pathogen system

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