Stochastic Lotka–Volterra food chains

Hening, Alexandru; Nguyen, Dang H.

doi:10.1007/s00285-017-1192-8

Cited by 50 publications

(35 citation statements)

References 47 publications

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“…These demographic effects include fluctuations in survival, growth, and reproduction and can result in fluctuations in population densities, disease prevalence, and genotypic frequencies. This stochasticity can drive populations extinct [Lewontin and Cohen, 1969, Gyllenberg et al, 1994, McLaughlin et al, 2002, Benaïm and Schreiber, 2009, Hening and Nguyen, 2018b, facilitate coexistence [Hutchinson, 1961, Chesson and Warner, 1981, Chesson, 1982, 1985, Chesson and Ellner, 1989, Chesson, 1994, Kuang and Chesson, 2009, Chesson, 2018, reverse competitive dominance [Benaïm and Lobry, 2016], maintain or disrupt genetic diversity [Gillespie, 1973, 1978, Gillespie and Turelli, 1989, and alter the persistence and spread of infectious diseases [Altizer et al, 2006]. One approach to studying these effects is to analyze stochastic difference or differential equations.…”

Section: Introductionmentioning

confidence: 99%

Persistence and extinction for stochastic ecological models with internal and external variables

2019

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The dynamics of species' densities depend both on internal and external variables. Internal variables include frequencies of individuals exhibiting different phenotypes or living in different spatial locations. External variables include abiotic factors or non-focal species. These internal or external variables may fluctuate due to stochastic fluctuations in environmental conditions. The interplay between these variables and species densities can determine whether a particular population persists or goes extinct. To understand this interplay, we prove theorems for stochastic persistence and exclusion for stochastic ecological difference equations accounting for internal and external variables. Specifically, we use a stochastic analog of average Lyapunov functions to develop sufficient and necessary conditions for (i) all population densities spending little time at low densities i.e. stochastic persistence, and (ii) population trajectories asymptotically approaching the extinction set with positive probability. For (i) and (ii), respectively, we provide quantitative estimates on the fraction of time that the system is near the extinction set, and the probability of asymptotic extinction as a function of the initial state of the system. Furthermore, in the case of persistence, we provide lower bounds for the expected time to escape neighborhoods of the extinction set. To illustrate the applicability of our results, we analyze stochastic models of evolutionary games, Lotka-Volterra dynamics, trait evolution, and spatially structured disease dynamics. Our analysis of these models demonstrates environmental stochasticity facilitates coexistence of strategies in the hawk-dove game, but inhibits coexistence in the rock-paper-scissors game and a Lotka-Volterra predator-prey model. Furthermore, environmental fluctuations with positive auto-correlations can promote persistence of evolving populations and persistence of diseases in patchy landscapes. While our results help close the gap between the persistence theories for deterministic and stochastic systems, we highlight several challenges for future research. 0 * Corresponding author: sschreiber@ucdavis.edu 1 arXiv:1808.07888v3 [q-bio.PE]

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Section: Introductionmentioning

confidence: 99%

Persistence and extinction for stochastic ecological models with internal and external variables

2019

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“…Hu et al [12] consider two correlative stochastic disturbances in the form of Gaussian white noise in an epidemic deterministic model constructed by Roberts and Jowett [13]. Also, Hening and Nguyen [14] construct a stochastic Lotka-Volterra food chain system by introducing n number of correlated Brownian motions into the deterministic food chain model. n is the total species number in the food chain and they use a coefficient matrix to convert the vector of correlated Brownian motions to a vector of independent standard Brownian motions.…”

Section: (T)mentioning

confidence: 99%

A stochastic differential equation SIS epidemic model with two correlated Brownian motions

Cai

Mao

2019

Nonlinear Dyn

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In this paper, we introduce two perturbations in the classical deterministic susceptible-infectedsusceptible epidemic model with two correlated Brownian motions. We consider two perturbations in the deterministic SIS model and formulate the original model as a stochastic differential equation with two correlated Brownian motions for the number of infected population, based on the previous work from Gray et al.

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“…The long‐time limit considers behavior of Φ for t → +∞ where

\frac{∂Φ}{\partial t} \to \overset{H}{true} = italicconst

. Here,

\overset{H}{true}

is referred to as the effective Hamiltonian, which is a value function under the temporal averaging in ergodic control . The limit equation for t → +∞ is

\overset{H}{true} + H ((), t, v, \frac{∂Φ}{\partial v}, \frac{\partial^{2} normalΦ}{\partial v^{2}}) = 0 in normalΩ,

where the left‐hand side is independent from t .…”

Section: Numerical Analysismentioning

confidence: 99%

Modeling stochastic operation of reservoir under ambiguity with an emphasis on river management

Yoshioka

2019

Optim Control Appl Methods

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An optimization problem of controlling a dam installed in a river is analyzed based on a stochastic control formalism of a diffusion process under model ambiguity: a new mathematical approach to this issue. The diffusion process is a pathwise unique solution to a water balance equation considering the inflow, outflow, water loss in the reservoir, and direct rainfall. Finding the optimal reservoir operation policy reduces to solving a degenerate parabolic partial differential equation: a Hamilton-Jacobi-Bellman-Isaacs equation. A monotone finite difference scheme is constructed for discretization of the equation, successfully generating nonoscillatory and reasonably accurate numerical solutions. Stability analysis of the resulting water balance dynamics is finally carried out for both environmentally friendly and not friendly reservoir operations. KEYWORDS finite difference scheme, Hamilton-Jacobi-Bellman-Isaacs equation, reservoir operation, stochastic control, viscosity solution INTRODUCTIONThis paper contributes to providing a mathematical framework for modeling and control of reservoirs created in rivers.Since the problem has an engineering background, it is first presented, and then, the issues to be addressed and our contributions are explained.Many rivers have dams serving as central infrastructures to supply essential water resources for human lives. 1-3 The stored water in the reservoir created by a dam is utilized for multiple purposes, such as drinking water, irrigation, and hydropower generation. 4 Several dams are used for flood mitigation as well. 5,6 On the other hand, controlling a dam critically affects its downstream river environment because the original flow regime is altered. 7-9 Sediment transport and attached algae population dynamics in dam downstream experience significant changes because dams trap sediment and the algae growth is highly flow-dependent. 10 Dam operation is thus desired to be environmentally friendly such that its impacts on the downstream are minimized, while it should be fit-for-purpose at the same time.Management of environment and ecology in rivers has been a long-standing engineering problem, 11-13 and mathematical tools and concepts for their modeling, analysis, and control are still developing. Several researchers considered river management problems based on stochastic process models such as stochastic differential equations (SDEs) 14 that can naturally consider stochasticity involved in environmental and ecological processes. 15 Interactions between aquatic species, such as riparian vegetation and fishes, and river discharges have been described with SDEs. 16 Miyamoto and Kimura 17 analyzed seedling, growth, and mortality dynamics of riparian trees subject to stochastic flood disturbances. Vesipa et al 18 764

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Stochastic Lotka–Volterra food chains

Cited by 50 publications

References 47 publications

Persistence and extinction for stochastic ecological models with internal and external variables

Persistence and extinction for stochastic ecological models with internal and external variables

A stochastic differential equation SIS epidemic model with two correlated Brownian motions

Modeling stochastic operation of reservoir under ambiguity with an emphasis on river management

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