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Discrete & Continuous Dynamical Systems - B

2015 , Volume 20 , Issue 6

Special issue on movement and dispersal in ecology, epidemiology and environmental science

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Preface
Robert Stephen Cantrell, Suzanne Lenhart, Yuan Lou and Shigui Ruan
2015, 20(6): i-iii doi: 10.3934/dcdsb.2015.20.6i +[Abstract](289) +[PDF](127.7KB)
Abstract:
The movement and dispersal of organisms have long been recognized as key components of ecological interactions and as such, they have figured prominently in mathematical models in ecology. More recently, dispersal has been recognized as an equally important consideration in epidemiology and in environmental science. Recognizing the increasing utility of employing mathematics to understand the role of movement and dispersal in ecology, epidemiology and environmental science, The University of Miami in December 2012 held a workshop entitled ``Everything Disperses to Miami: The Role of Movement and Dispersal in Ecology, Epidemiology and Environmental Science" (EDM).

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Extinction in discrete, competitive, multi-species patch models
David M. Chan, Matt McCombs, Sarah Boegner, Hye Jin Ban and Suzanne L. Robertson
2015, 20(6): 1583-1590 doi: 10.3934/dcdsb.2015.20.1583 +[Abstract](345) +[PDF](322.4KB)
Abstract:
In this paper we extend the results of Franke and Yakubu in [5] for extinction in discrete competitive patch models. For a system of $n$ species on $m$ patches, we define conditions under which one species is a ``superior competitor" to another and show that this is sufficient for one species to drive another to extinction. We also illustrate the result with an example for three species on three patches.
Spatial population dynamics in a producer-scrounger model
Chris Cosner and Andrew L. Nevai
2015, 20(6): 1591-1607 doi: 10.3934/dcdsb.2015.20.1591 +[Abstract](387) +[PDF](614.6KB)
Abstract:
The spatial population dynamics of an ecological system involving producers and scroungers is studied using a reaction-diffusion model. The two populations move randomly and increase logistically, with birth rates determined by the amount of resource acquired. Producers can obtain the resource directly from the environment, but must surrender a proportion of their discoveries to nearby scroungers through a process known as scramble kleptoparasitism. The proportion of resources stolen by a scrounger from nearby producers decreases as the local scrounger density increases. Parameter combinations which allow producers and scroungers to persist either alone or together are distinguished from those in which they cannot. Producer persistence depends in general on the distribution of resources and producer movement, whereas scrounger persistence depends on its ability to invade when producers are at steady-state. It is found that (i) both species can persist when the habitat has high productivity, (ii) neither species can persist when the habitat has low productivity, and (iii) slower dispersal of both the producer and scrounger is favored when the habitat has intermediate productivity.
Partial differential equations with Robin boundary condition in online social networks
Guowei Dai, Ruyun Ma, Haiyan Wang, Feng Wang and Kuai Xu
2015, 20(6): 1609-1624 doi: 10.3934/dcdsb.2015.20.1609 +[Abstract](1394) +[PDF](303.4KB)
Abstract:
In recent years, online social networks such as Twitter, have become a major source of information exchange and research on information diffusion in social networks has been accelerated. Partial differential equations are proposed to characterize temporal and spatial patterns of information diffusion over online social networks. The new modeling approach presents a new analytic framework towards quantifying information diffusion through the interplay of structural and topical influences. In this paper we develop a non-autonomous diffusive logistic model with indefinite weight and the Robin boundary condition to describe information diffusion in online social networks. It is validated with a real dataset from an online social network, Digg.com. The simulation shows that the logistic model with the Robin boundary condition is able to more accurately predict the density of influenced users. We study the bifurcation, stability of the diffusive logistic model with heterogeneity in distance. The bifurcation and stability results of the model information describe either information spreading or vanishing in online social networks.
Mathematical study of the effects of travel costs on optimal dispersal in a two-patch model
Theodore E. Galanthay
2015, 20(6): 1625-1638 doi: 10.3934/dcdsb.2015.20.1625 +[Abstract](264) +[PDF](378.9KB)
Abstract:
The theoretical dispersal of organisms has been widely studied. It is well known for single species dispersal in a spatially heterogeneous and temporally constant environment that ``balanced dispersal'' is an evolutionarily stable strategy [36,10]. This assumes that organisms do not pay a cost to move from one part of the environment to another. We begin this paper by proving that the optimal strategy for organisms constrained by perceptual limitations, described by [19], is evolutionarily stable. Then, we extend this idea of optimal dispersal to a situation where constrained organisms pay a cost to move between two patches in a heterogeneous environment. For moderate travel costs, we find a convergent stable strategy that suggests an extension of the balanced dispersal concept. Furthermore, we show for high costs that the best strategy is to ignore information about the environment.
Time-invariant and stochastic disperser-structured matrix models: Invasion rates of fleshy-fruited exotic shrubs
Carol C. Horvitz, Anthony L. Koop and Kelley D. Erickson
2015, 20(6): 1639-1662 doi: 10.3934/dcdsb.2015.20.1639 +[Abstract](313) +[PDF](1234.5KB)
Abstract:
Interest in spatial population dynamics includes applications to the spread of disease and invasive species. Recently, models for structured populations have been extended to incorporate temporal variation in both demography and dispersal. Here we propose a novel version of the model that incorporates structured dispersal to evaluate how changes in the relative proportion of mammalian, and short- and long-distance avian dispersers affect the rate of spread of an invasive shrub, Ardisia elliptica in Everglades National Park. We implemented $45$ time-invariant models, including one in which a single dispersal kernel was estimated from field data by pooling all seedlings, and $44$ that were disperser-structured in which dispersal kernels were estimated separately for gravity-, catbird-, robin- and raccoon-dispersed seed. Robins, the longest distance dispersers, are infrequent. Finally we implemented a time-varying model that included variability among years in the proportion of seeds that were taken by robins. The models estimated invasion speeds that ranged from $3.9$ to $34.7$ m $ yr^{-1}$ . Infrequent long-distance dispersal by robins were important in determining invasion speed in the disperser-structured model. Comparing model projections with the (historically) known rate of spread, we show how a model that stratifies seeds by dispersal agents does better than one that ignores them, although all of our models underestimate it.
Spreading speeds and traveling wave solutions in cooperative integral-differential systems
Changbing Hu, Yang Kuang, Bingtuan Li and Hao Liu
2015, 20(6): 1663-1684 doi: 10.3934/dcdsb.2015.20.1663 +[Abstract](302) +[PDF](562.8KB)
Abstract:
We study a cooperative system of integro-differential equations. It is shown that the system in general has multiple spreading speeds, and when the linear determinacy conditions are satisfied all the spreading speeds are the same and equal to the spreading speed of the linearized system. The existence of traveling wave solutions is established via integral systems. It is shown that when the linear determinacy conditions are satisfied, if the unique spreading speed is not zero then it may be characterized as the slowest speed of a class of traveling wave solutions. Some examples are presented to illustrate the theoretical results.
Modeling of contact tracing in epidemic populations structured by disease age
Xi Huo
2015, 20(6): 1685-1713 doi: 10.3934/dcdsb.2015.20.1685 +[Abstract](347) +[PDF](968.5KB)
Abstract:
We consider an age-structured epidemic model with two basic public health interventions: (i) identifying and isolating symptomatic cases, and (ii) tracing and quarantine of the contacts of identified infectives. The dynamics of the infected population are modeled by a nonlinear infection-age-dependent partial differential equation, which is coupled with an ordinary differential equation that describes the dynamics of the susceptible population. Theoretical results about global existence and uniqueness of positive solutions are proved. We also present two practical applications of our model: (1) we assess public health guidelines about emergency preparedness and response in the event of a smallpox bioterrorist attack; (2) we simulate the 2003 SARS outbreak in Taiwan and estimate the number of cases avoided by contact tracing. Our model can be applied as a rational basis for decision makers to guide interventions and deploy public health resources in future epidemics.
Asymptotical behaviors of a general diffusive consumer-resource model with maturation delay
Wonlyul Ko, Inkyung Ahn and Shengqiang Liu
2015, 20(6): 1715-1733 doi: 10.3934/dcdsb.2015.20.1715 +[Abstract](336) +[PDF](485.5KB)
Abstract:
In this paper, we examine the asymptotic behaviors of a diffusive delayed consumer-resource model subject to homogeneous Neumann boundary conditions, where the discrete time delay covers the period from the birth of juvenile consumers to their maturity, and the predation is of a general type of functional response. We construct the threshold dynamics of the persistence and extinction of the consumer. Moreover, we establish the sufficient conditions for the global attractivity of the semitrivial and coexistence equilibria. Finally, we apply our results to the specific consumer-resource models with Beddington-DeAngelis, Crowley-Martin, and ratio-dependent type of functional responses.
Hopf bifurcation for a spatially and age structured population dynamics model
Zhihua Liu, Hui Tang and Pierre Magal
2015, 20(6): 1735-1757 doi: 10.3934/dcdsb.2015.20.1735 +[Abstract](360) +[PDF](825.4KB)
Abstract:
This paper is devoted to the study of a spatially and age structured population dynamics model. We study the stability and Hopf bifurcation of the positive equilibrium of the model by using a bifurcation theory in the context of integrated semigroups. This problem is a first example for Hopf bifurcation for a spatially and age/size structured population dynamics model. Bifurcation analysis indicates that Hopf bifurcation occurs at a positive age/size dependent steady state of the model. The results are confirmed by some numerical simulations.
Optimal control of integrodifference equations in a pest-pathogen system
Marco V. Martinez, Suzanne Lenhart and K. A. Jane White
2015, 20(6): 1759-1783 doi: 10.3934/dcdsb.2015.20.1759 +[Abstract](689) +[PDF](3533.2KB)
Abstract:
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 forward-backward sweep numerical method to implement our optimization which produces spatio-temporal control strategies for the gypsy moth case study.
How does the spreading speed associated with the Fisher-KPP equation depend on random stationary diffusion and reaction terms?
Gregoire Nadin
2015, 20(6): 1785-1803 doi: 10.3934/dcdsb.2015.20.1785 +[Abstract](339) +[PDF](504.3KB)
Abstract:
We consider one-dimensional reaction-diffusion equations of Fisher-KPP type with random stationary ergodic coefficients. A classical result of Freidlin and Gartner [16] yields that the solutions of the initial value problems associated with compactly supported initial data admit a linear spreading speed almost surely. We use in this paper a new characterization of this spreading speed recently proved in [8] in order to investigate the dependence of this speed with respect to the heterogeneity of the diffusion and reaction terms. We prove in particular that adding a reaction term with null average or rescaling the coefficients by the change of variables $x\to x/L$, with $L>1$, speeds up the propagation. From a modelling point of view, these results mean that adding some heterogeneity in the medium gives a higher invasion speed, while fragmentation of the medium slows down the invasion.
Positive steady state solutions of a plant-pollinator model with diffusion
Lijuan Wang, Hongling Jiang and Ying Li
2015, 20(6): 1805-1819 doi: 10.3934/dcdsb.2015.20.1805 +[Abstract](315) +[PDF](829.0KB)
Abstract:
In this paper, a plant-pollinator population system with diffusion is investigated, which is described by a cooperative model with B-D functional response. Using the Leray-Schauder degree theory, we discuss the existence of positive steady state solutions of the model. The result shows when the growth rate of plants is large and the death rate of pollinators is small, the plants and pollinators can coexist. By the regular perturbation theorem and monotone dynamical system theory, the uniqueness and stability of positive solutions have been studied. Especially, we show that the unique positive solution is a global attractor under some conditions. Furthermore, we present some numerical simulations, which is not only to check our theoretical results but also to supply some conjectures out of theoretical analysis.
Transversality for time-periodic competitive-cooperative tridiagonal systems
Yi Wang and Dun Zhou
2015, 20(6): 1821-1830 doi: 10.3934/dcdsb.2015.20.1821 +[Abstract](318) +[PDF](357.6KB)
Abstract:
Transversality of the stable and unstable manifolds of hyperbolic periodic solutions is proved for tridiagonal competitive-cooperative time-periodic systems. We further show that such systems admit the Morse-Smale property provided that all the fixed points (of the corresponding Poincaré map) are hyperbolic. The main tools used here are the integer-valued Lyapunov function, as well as the Floquet theory developed in [1] for general time-dependent tridiagonal linear systems.
Spatial dynamics of a diffusive predator-prey model with stage structure
Liang Zhang and Zhi-Cheng Wang
2015, 20(6): 1831-1853 doi: 10.3934/dcdsb.2015.20.1831 +[Abstract](333) +[PDF](483.1KB)
Abstract:
In this paper, we propose a nonlocal and time-delayed reaction-diffusion predator-prey model with stage structure. It is assumed that prey individuals undergo two stages, immature and mature, and the conversion of consumed prey biomass into predator biomass has a retardation. In terms of the principal eigenvalue of nonlocal elliptic eigenvalue problems, we establish the uniform persistence and global extinction for the model. In particular, the uniform persistence implies the existence of positive steady states. Finally, we investigate a specially spatially homogeneous predator-prey system and show the complicated dynamics of the system due to the non-local delay in the prey equation.

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