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1531-3492

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1553-524X

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### Volume 10, 2008

## Discrete & Continuous Dynamical Systems - B

2014 , Volume 19 , Issue 10

Special issue in honor of Chris Cosner on the occasion of his 60th birthday

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2014, 19(10): i-ii
doi: 10.3934/dcdsb.2014.19.1i

*+*[Abstract](242)*+*[PDF](81.9KB)**Abstract:**

Chris Cosner turned 60 on June 3, 2012 and now, at age 62, continues his stellar career at the interface of mathematics and biology. He received his Ph.D. in 1977 at the University of California, Berkeley under the direction of Murray Protter, winning the Bernard Friedman prize for the best dissertation in applied mathematics. From 1977 until 1982 he was on the faculty of Texas A&M University. In 1982 he left A&M to join the faculty of the Department of Mathematics of the University of Miami as Associate Professor, rising to the rank of Professor in 1988. The academic year 2013-2014 marked his 32nd year of distinguished service to the University of Miami and its research and pedagogical missions.

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2014, 19(10): 3031-3056
doi: 10.3934/dcdsb.2014.19.3031

*+*[Abstract](143)*+*[PDF](538.7KB)**Abstract:**

In this paper we refine in a substantial way part of the materials of the celebrated paper of Belgacem and Cosner [3] by considering a rather general class of degenerate diffusive logistic equations in the presence of advection. Rather paradoxically, a large advection can provoke the stabilization to an steady state of a former explosive solution. Similarly, even with a severe taxis down the environmental gradient the species can be permanent.

2014, 19(10): 3057-3085
doi: 10.3934/dcdsb.2014.19.3057

*+*[Abstract](286)*+*[PDF](1293.3KB)**Abstract:**

In this paper, we investigate the inside dynamics of the positive solutions of integro-differential equations \begin{equation*} \partial_t u(t,x)= (J\star u)(t,x) -u(t,x) + f(u(t,x)), \ t>0 \hbox{ and } x\in\mathbb{R}, \end{equation*} with both

*thin-tailed*and

*fat-tailed*dispersal kernels $J$ and a monostable reaction term $f.$ The notion of inside dynamics has been introduced to characterize traveling waves of some reaction-diffusion equations [23]. Assuming that the solution is made of several fractions $\upsilon^i\ge 0$ ($i\in I \subset \mathbb{N}$), its inside dynamics is given by the spatio-temporal evolution of $\upsilon^i$. According to this dynamics, the traveling waves can be classified in two categories: pushed and pulled waves. For thin-tailed kernels, we observe no qualitative differences between the traveling waves of the above integro-differential equations and the traveling waves of the classical reaction-diffusion equations. In particular, in the KPP case ($f(u)\leq f'(0)u$ for all $u\in(0,1)$) we prove that all the traveling waves are pulled. On the other hand for fat-tailed kernels, the integro-differential equations do not admit any traveling waves. Therefore, to analyse the inside dynamics of a solution in this case, we introduce new notions of pulled and pushed solutions. Within this new framework, we provide analytical and numerical results showing that the solutions of integro-differential equations involving a fat-tailed dispersal kernel are pushed. Our results have applications in population genetics. They show that the existence of long distance dispersal events during a colonization tend to preserve the genetic diversity.

2014, 19(10): 3087-3104
doi: 10.3934/dcdsb.2014.19.3087

*+*[Abstract](224)*+*[PDF](836.0KB)**Abstract:**

Partial differential equation models of diffusion and advection are fundamental to understanding population behavior and interactions in space, but can be difficult to analyze when space is heterogeneous. As a proxy for partial differential equation models, and to provide some insight into a few questions regarding growth and movement patterns of a single population and two competing populations, a simple three-patch system is used. For a single population it is shown that diffusion rates occur for which the total biomass supported on a heterogeneous landscape exceeds total carrying capacity, confirming previous studies of partial differential equations and other models. It is also shown that the total population supported can increase indefinitely as the sharpness of the heterogeneity increases. For two competing species, it is shown that adding advection to a reaction-diffusion system can potentially reverse the general rule that the species with smaller diffusion rates always wins, or lead to coexistence. Competitive dominance is also favored for the species for which the sharpness of spatial heterogeneity in growth rate is greater. The results are consistent with analyses of partial differential equations, but the patch approach has some advantages in being more intuitively understandable.

2014, 19(10): 3105-3132
doi: 10.3934/dcdsb.2014.19.3105

*+*[Abstract](238)*+*[PDF](506.3KB)**Abstract:**

In this paper we consider the diffusive competition model consisting of an invasive species with density $u$ and a native species with density $v$, in a radially symmetric setting with free boundary. We assume that $v$ undergoes diffusion and growth in $\mathbb{R}^N$, and $u$ exists initially in a ball ${r < h(0)}$, but invades into the environment with spreading front ${r = h(t)}$, with $h(t)$ evolving according to the free boundary condition $h'(t) = -\mu u_r(t, h(t))$, where $\mu>0$ is a given constant and $u(t,h(t))=0$. Thus the population range of $u$ is the expanding ball ${r < h(t)}$, while that for $v$ is $\mathbb{R}^N$. In the case that $u$ is a superior competitor (determined by the reaction terms), we show that a spreading-vanishing dichotomy holds, namely, as $t\to\infty$, either $h(t)\to\infty$ and $(u,v)\to (u^*,0)$, or $\lim_{t\to\infty} h(t)<\infty$ and $(u,v)\to (0,v^*)$, where $(u^*,0)$ and $(0, v^*)$ are the semitrivial steady-states of the system. Moreover, when spreading of $u$ happens, some rough estimates of the spreading speed are also given. When $u$ is an inferior competitor, we show that $(u,v)\to (0,v^*)$ as $t\to\infty$, so the invasive species $u$ always vanishes in the long run.

2014, 19(10): 3133-3145
doi: 10.3934/dcdsb.2014.19.3133

*+*[Abstract](176)*+*[PDF](480.2KB)**Abstract:**

Based on the classical Ross-Macdonald model, in this paper we propose a periodic malaria model to incorporate the effects of temporal and spatial heterogeneity on disease transmission. The temporal heterogeneity is described by assuming that some model coefficients are time-periodic, while the spatial heterogeneity is modeled by using a multi-patch structure and assuming that individuals travel among patches. We calculate the basic reproduction number $\mathcal{R}_0$ and show that either the disease-free periodic solution is globally asymptotically stable if $\mathcal{R}_0\le 1$ or the positive periodic solution is globally asymptotically stable if $\mathcal{R}_0>1$. Numerical simulations are conducted to confirm the analytical results and explore the effect of travel control on the disease prevalence.

2014, 19(10): 3147-3167
doi: 10.3934/dcdsb.2014.19.3147

*+*[Abstract](187)*+*[PDF](803.4KB)**Abstract:**

Recent studies have suggested that the risk of exposure to Lyme disease is emerging in Canada because of the expanding range of

*I. scapularis*ticks. The wide geographic breeding range of

*I. scapularis*-carrying migratory birds is consistent with the widespread geographical occurrence of

*I. scapularis*in Canada. However, how important migratory birds from the United States are for the establishment and the stable endemic transmission cycle of Lyme disease in Canada remains an issue of theoretical challenge and practical significance. In this paper, we design and analyze a periodic model of differential equations with a forcing term modeling the annual bird migration to address the aforementioned issue. Our results show that ticks can establish in any migratory bird stopovers and breeding sites. Moreover, bird-transported ticks may increase the probability of

*B. burgdorferi*establishment in a tick-endemic habitat.

2014, 19(10): 3169-3189
doi: 10.3934/dcdsb.2014.19.3169

*+*[Abstract](184)*+*[PDF](506.9KB)**Abstract:**

The dynamics of a reaction-diffusion system for two species of microorganism in an unstirred chemostat with internal storage is studied. It is shown that the diffusion coefficient is a key parameter of determining the asymptotic dynamics, and there exists a threshold diffusion coefficient above which both species become extinct. On the other hand, for diffusion coefficient below the threshold, either one species or both species persist, and in the asymptotic limit, a steady state showing competition exclusion or coexistence is reached.

2014, 19(10): 3191-3207
doi: 10.3934/dcdsb.2014.19.3191

*+*[Abstract](176)*+*[PDF](566.7KB)**Abstract:**

The population dynamics of species with separate growth and dispersal stages can be modeled by a discrete-time, continuous-space integrodifference equation. Many authors have considered the case where the model parameters remain fixed over time, however real environments are constantly in flux. We develop a framework for analyzing the population dynamics when the dispersal parameters change over time in a cyclic fashion. In particular, for the case of $N$ cyclic dispersal kernels modeling movement in the presence of unidirectional flow, we derive a $2N^{th}$-order boundary value problem that can be used to study the linear stability of the associated integrodifference model.

2014, 19(10): 3209-3218
doi: 10.3934/dcdsb.2014.19.3209

*+*[Abstract](193)*+*[PDF](326.3KB)**Abstract:**

A model is considered for a spatially distributed population of male and female individuals that mate and reproduce only once in their life during a very short reproductive season. Between birth and mating, females and males move by diffusion on a bounded domain $\Omega$. Mating and reproduction is described by a (positively) homogeneous function (of degree one). We identify a basic reproduction number $\mathcal{R}_0$ that acts as a threshold between extinction and persistence. If $\mathcal{R}_0 <1$, the population dies out while it persists (uniformly weakly) if $\mathcal{R}_0 > 1$. $\mathcal{R}_0$ is the cone spectral radius of a bounded homogeneous map.

2014, 19(10): 3219-3244
doi: 10.3934/dcdsb.2014.19.3219

*+*[Abstract](227)*+*[PDF](607.4KB)**Abstract:**

Recently, the ideal free dispersal strategy has been proven to be evolutionarily stable in the spatially discrete as well as continuous setting. That is, at equilibrium a species adopting the strategy is immune against invasion by any species carrying a different dispersal strategy, other conditions being held equal. In this paper, we consider a two-species competition model where one of the species adopts an ideal free dispersal strategy, but is penalized by a weak Allee effect. We will show rigorously in this case that the ideal free disperser is invasible by a range of non-ideal free strategies, illustrating the trade-off between the advantage of being an ideal free disperser and the setback caused by the weak Allee effect. Moreover, an integral criterion is given to determine the stability/instability of one of the semi-trivial steady states, which is always linearly neutrally stable due to the degeneracy caused by the weak Allee effect.

2014, 19(10): 3245-3265
doi: 10.3934/dcdsb.2014.19.3245

*+*[Abstract](185)*+*[PDF](452.8KB)**Abstract:**

This paper studies Holder continuity of weak solutions to strongly coupled elliptic systems. We do not assume that the solutions are bounded but BMO and the ellipticity constants can be unbounded.

2014, 19(10): 3267-3281
doi: 10.3934/dcdsb.2014.19.3267

*+*[Abstract](288)*+*[PDF](537.2KB)**Abstract:**

We propose a reaction-advection-diffusion model to study competition between two species in a stream. We divide each species into two compartments, individuals inhabiting the benthos and individuals drifting in the stream. We assume that the growth of and competitive interactions between the populations take place on the benthos and that dispersal occurs in the stream. Our system consists of two linear reaction-advection-diffusion equations and two ordinary differential equations. Here, we provide a thorough study for the corresponding single species model, which has been previously proposed. We next give formulas for the rightward spreading and leftward spreading speed for the model. We show that rightward spreading speed can be characterized as is the slowest speed of a class of traveling wave speeds. We provide sharp conditions for the spreading speeds to be positive. For the two species competition model, we investigate how a species spreads into its competitor's environment. Formulas for the spreading speeds are provided under linear determinacy conditions. We demonstrate that under certain conditions, the invading species can spread upstream. Lastly, we study the existence of traveling wave solutions for the two species competition model.

2014, 19(10): 3283-3298
doi: 10.3934/dcdsb.2014.19.3283

*+*[Abstract](282)*+*[PDF](592.9KB)**Abstract:**

We present a mathematical model for the localised control of mosquitoes using larvivorous fish. It is supposed that the adult mosquitoes choose among a finite number of isolated ponds for oviposition and that these ponds differ in various respects including physical size, survival prospects and maturation times for mosquito larvae. We model a mosquito control effort that involves stocking some or all of these ponds with larvivorous fish such as the mosquitofish

*Gambusia affinis*. The effect of doing so may vary from pond to pond, and the ponds are coupled via the adult mosquitoes in the air. Also, adult mosquitoes may avoid ovipositing in ponds containing the larvivorous fish. Our model enables us to predict how the larvivorous fish should be allocated between ponds, and shows in particular that only certain ponds should be stocked if there is a limited supply of the fish. We also consider oviposition pond selection by mosquitoes, and show that in some situations mosquitoes might do better to simply choose a pond at random.

2014, 19(10): 3299-3317
doi: 10.3934/dcdsb.2014.19.3299

*+*[Abstract](147)*+*[PDF](1008.3KB)**Abstract:**

We present a mathematical model for a technology cycle that centers its attention on the coexistence mechanisms of competing technologies. We use a biological analogy to couple the adoption of a technology with the provision of financial resources. In our model financial resources are limited and provided at a constant rate. There are two variants analyzed in this work, the first considers the so-called internal innovation and the second introduces external innovation. We make use of the adaptive dynamics framework to explain the persistence of closely related technologies as opposed to the usual competitive exclusion of all but one dominant technology. For internal innovation the existence of a resource remanent in the full adoption case does not always lead to competitive exclusion; otherwise with the external innovation the resident technology can not be displaced. The paper illustrates the persistence of closely related technologies and the competitive exclusion in renewable energy technologies and TV sets respectively.

2014, 19(10): 3319-3340
doi: 10.3934/dcdsb.2014.19.3319

*+*[Abstract](147)*+*[PDF](752.4KB)**Abstract:**

We use an evolutionary approach to find ``most appropriate'' dispersal models for ecological applications. From a random walk with locally or nonlocally defined transition probabilities we derive a family of diffusion equations. We assume a monotonic dependence of its diffusion coefficient on the local population fitness and search for a model within this class that can invade populations with other dispersal type from the same class but is not invadable itself. We propose an optimization technique using numerically obtained principal eigenvalue of the invasion problem and obtain two candidates for evolutionary stable dispersal strategy: Fokker-Planck equation with diffusion coefficient decreasing with fitness and Attractive Diffusion equation (Okubo and Levin, 2001) with diffusion coefficient increasing with fitness. For FP case the transition probabilities are defined by the departure point and for AD case by the destination point. We show that for the case of small spatial variability of the population growth rate both models are close to the model for ideal free distribution by Cantrell et al. (2008).

2014, 19(10): 3341-3357
doi: 10.3934/dcdsb.2014.19.3341

*+*[Abstract](169)*+*[PDF](442.8KB)**Abstract:**

Recent experimental studies have shown that HIV can be transmitted directly from cell to cell when structures called virological synapses form during interactions between T cells. In this article, we describe a new within-host model of HIV infection that incorporates two mechanisms: infection by free virions and the direct cell-to-cell transmission. We conduct the local and global stability analysis of the model. We show that if the basic reproduction number ${\mathcal R}_0\leq 1$, the virus is cleared and the disease dies out; if ${\mathcal R}_0>1$, the virus persists in the host. We also prove that the unique positive equilibrium attracts all positive solutions under additional assumptions on the parameters. Finally, a multi strain model incorporating cell-to-cell viral transmission is proposed and shown to exhibit a competitive exclusion principle.

2014, 19(10): 3359-3378
doi: 10.3934/dcdsb.2014.19.3359

*+*[Abstract](216)*+*[PDF](957.4KB)**Abstract:**

Mass migrations of vertebrate and arthropod species have long been perceived as some of the most mystical phenomena in nature. And for eons, we have been asking ourselves why animals migrate. Ecologically, migration provides benefits in currencies of survival, growth, and reproduction, allowing animals to exploit environmental heterogeneities in space and time. Yet for a given environment, different species respond with different behaviors -- some travelling large distances, while others shelter in place. Part of the explanation of this distinction is the physiological differences between species and their ability to move. But is physiological difference a necessary pre-condition? Or can environmental heterogeneity itself be sufficient for bifurcations in movement behavior?

In this paper, we address this last question using a model for the evolution of migration in a density-independent, spatially-explicit setting when movement is costly based on the harvesting a single resource that varies in space and time. We use optimal control methods to calculate the optimal movement patterns in several different situations. In this framework, optimal movement strategies can be classified into six different regimes, based on the cost of movement, the strength and scale of seasonal resource variation, and the degree of trade-off between short-term and long-term benefits. We show that a migratory niche emerges in response to inseparable spatio-temporal environmental heterogeneity, and that this niche can bifurcate from changes to the resource distribution without need for physiological divergence.

2014, 19(10): 3379-3396
doi: 10.3934/dcdsb.2014.19.3379

*+*[Abstract](164)*+*[PDF](644.0KB)**Abstract:**

To study the effect of immune response in viral infections, a new mathematical model is proposed and analyzed. It describes the interactions between susceptible host cells, infected host cells, free virus, lytic and nonlytic immune response. Using the LaSalle's invariance principle, we establish conditions for the global stability of equilibria. Uniform persistence is obtained when there is a unique endemic equilibrium. Mathematical analysis and numerical simulations indicate that the basic reproduction number of the virus and immune response reproductive number are sharp threshold parameters to determine outcomes of infection. Lytic and nonlytic antiviral activities play a significant role in the amount of susceptible host cells and immune cells in the endemic steady state. We also present potential applications of the model in clinical practice by introducing antiviral effects of antiviral drugs.

2014, 19(10): 3397-3432
doi: 10.3934/dcdsb.2014.19.3397

*+*[Abstract](164)*+*[PDF](3194.4KB)**Abstract:**

We investigate the dynamics of a predator-prey system with the assumption that both prey and predators use game theory-based strategies to maximize their per capita population growth rates. The predators adjust their strategies in order to catch more prey per unit time, while the prey, on the other hand, adjust their reactions to minimize the chances of being caught. We assume each individual is either mobile or sessile and investigate the evolution of mobility for each species in the predator-prey system. When the underlying population dynamics is of the Lotka-Volterra type, we show that strategies evolve to the equilibrium predicted by evolutionary game theory and that population sizes approach their corresponding stable equilibrium (i.e. strategy and population effects can be analyzed separately). This is no longer the case when population dynamics is based on the Holling II functional response, although the strategic analysis still provides a valuable intuition into the long term outcome. Numerical simulation results indicate that, for some parameter values, the system has chaotic behavior. Our investigation reveals the relationship between the game theory-based reactions of prey and predators, and their population changes.

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