# 46th Annual John H. Barrett Memorial Lectures

## Modeling and Analysis of Nonlinear PDEs in Spatial Ecology

### Plenary Speakers:

Chris Cosner, University of Miami, Coral Gables, FL
Mark Lewis, University of Alberta, Edmonton, Alberta, Canada [website]
Yuan Lou, The Ohio State University [website]

### Invited Speakers:

Anna R. Ghazaryan, Miami University, Oxford, OH [website]
Yu Jin, University of Nebraska, NE [website]
Judith Miller, Georgetown University, DC [website]
Nancy Rodriguez, University of North Carolina, Chapel Hill, NC [website]
Wenxian Shen, Auburn University, AL [website]
Michael Winkler, University of Paderborn, Germany [website]

#### Organizing Committee:

• Louis J. Gross
• Suzanne Lenhart
• Tuoc Phan

### Plenary Speakers:

Chris Cosner
University of Miami, Coral Gables, FL
Talk 1: The reduction principle, the ideal free distribution, and the evolution of dispersal strategies
The problem of understanding the evolution of dispersal has attracted much attention from mathematicians and biologists in recent years. For reaction-diffusion models and their nonlocal and discrete analogues, in environments that vary in space but not in time, the strategy of not dispersing at all is often convergence stable within in many classes of strategies. This is related to a “reduction principle” which states that that in general dispersal reduces population growth rates. However, when the class of feasible strategies includes strategies that generate an ideal free population distribution at equilibrium (all individuals have equal fitness, with no net movement), such strategies are known to be evolutionarily stable in various cases. Much of the work in this area involves using ideas from dynamical systems theory and partial differential equations to analyze pairwise invasibility problems, which are motivated by ideas from adaptive dynamics and ultimately game theory. The talk will describe some past results and current work on these topics.

Talk 2: Spatial population models with fitness based dispersal
Traditional continuous time models in spatial ecology typically describe movement in terms of linear diffusion and advection, which combine with nonlinear population dynamics to produce semilinear equations and systems. However, if organisms are assumed to move up gradients of their reproductive fitness, and fitness is density dependent (for example logistic), the resulting models are quasilinear and may have other novel features. This talk will describe some models involving fitness dependent dispersal and some results and challenges in the analysis of such models.

Mark Lewis
University of Alberta
Talk 1: Genetic consequences of range expansion under climate change
Range expansion is a crucial population response to climate change. The genetic consequences are not well understood but are clearly coupled to ecological dynamics that, in turn, are driven by shifting climate conditions. We model a population with a reaction–diffusion system, coupled to a heterogeneous environment that shifts with time due to climate change. We decompose the resulting traveling wave solution into neutral genetic components to analyze the spatio-temporal dynamics of its genetic structure. Our analysis shows that range expansion under slow climate change preserves genetic diversity. However, diversity is diminished when the climate change occurs too quickly. We show that populations with intermediate dispersal ability are best for maintaining genetic diversity during shifting climatic conditions. Our study also provides new analytical insight regarding dynamics of traveling wave solutions in heterogeneous environments. This work is joint with Jimmy Garnier (CNRS).

Talk 2: Mathematics behind stream population dynamics
Human activities change the natural flow regimes in streams and rivers and this impacts ecosystems. In this talk I will mathematically investigate the impact of changes in water flow on biological populations. The approach I will take is to develop process-oriented advection-diffusion-reaction equations that couple hydraulic flow to population growth, and then to analyze the equations so as to assess the effect of impacts of water flow on population dynamics. The mathematical framework is based on new theory for the net reproductive rate Ro as applied to advection-diffusion-reaction equations. I will then connect the theory to populations in rivers under various flow regimes.
This work lays the groundwork for connecting Ro to more complex models of spatially structured and interacting populations, as well as more detailed habitat and hydrological data. This is achieved through explicit numerical simulation of two dimensional depth-averaged models for river population dynamics.

Yuan Lou
The Ohio State University
Talk 1: Dispersal in advective environments
We consider some mathematical models in advective environments, where individuals are exposed to unidirectional flow, with the possibility of being lost through the boundary. We study the persistence and range for a single species. We also consider the evolution of dispersal in such advective environments. Our analysis suggests that, in contrast to the case of no advection, slow dispersal is generally selected against in advective environments, and fast or intermediate dispersal rate will be favored. This talk is based on joint works wth King-Yeung Lam, Frithjof Lutscher, Peng Zhou and Dongmei Xiao.

Talk 2: Evolution of diffusion in a mutation-selection model
We consider a mutation-selection model of a population structured by the spatial variables and a trait variable which is the diffusion rate. Competition for resource is local in spatial variables, but nonlocal in trait. We show that in the limit of small mutation rate, steady state solutions remain regular in the spatial variables and yet concentrates in the trait variable and forms a Dirac mass supported at the lowest diffusion rate. This is a joint work with King-Yeung Lam.

### Invited Speakers:

Anna Ghazaryan
Miami University, Oxford, OH
Traveling Fronts in Holling-Tanner Model with Slow Diffusion
For wide range of parameters, we study traveling waves in a diffusive version of the Holling-Tanner predator-prey model. Frontwaves are constructed using geometric singular perturbation theory and the theory of rotated vector fields. We focus on the appearance of the fronts in various singular limits.

Judith Miller
Georgetown University, DC
Invasion slowing and pinning due to spatially heterogeneous selection
In 1997, Kirkpatrick and Barton developed and numerically solved a partial differential equation model for the joint evolution of population density and the mean of a quantitative trait in time and space. While enjoying high interest among biologists, the Kirkpatrick-Barton system has resisted rigorous analysis. We prove existence of both pinned states and traveling waves for this under certain scalings, for small positive values of two derived parameters. These model “genetic swamping” and invasions in the presence of a spatial trait optimum gradient. We also examine the effect of maladaptation on invasion speeds.

Nancy Rodriguez
University of North Carolina, Chapel Hill, NC
On reaction-advection-diffusion models for multi-species segregation
The inclusion of cross-diffusion in a model is important when trying to understand the dynamics of a multi-species population where the populations try to avoid each other. Such models have been proposed since the 50’s and its effects have been analyzed vastly in the literature. Cross-diffusion can lead to many interesting patterns and generally makes it much harder to answer fundamental questions, such as the global well-posedness and regularity of the system. We discuss a system that was derived as a particle model for population segregation. Previous numerical results have demonstrated both strong and weak-segregation. In this talk, I will discuss the existence of steady-state solutions and mention the effect that the environment and various mobilities (how easily a population can navigate the environment) has on the qualitative behavior of the solution.
Furthermore, we discuss the global well-posedness for the system pointing out the main difficulties.

Wenxian Shen
Auburn University, AL
Principal Spectral Theory of Nonlocal Dispersal Operators with Time Periodic Indefinite Weight Functions and Applications
In the current talk, I will present some principal spectral theory for nonlocal dispersal operators with time periodic inde nite weight functions subject to Dirichlet type, Neumann type, or spatially periodic boundary conditions. I will rst review some existing principal spectral theory for nonlocal dispersal operators with time periodic dependence in the regular sense. Next I will consider necessary and sufficient conditions for the existence of a unique positive principal spectrum point for nonlocal dispersal operators with time periodic inde nite weight functions. Finally I will discuss some applications to nonlinear mathematical models from biology.

Michael Winkler
Mathematical challenges in the analysis of chemotaxis-fluid interaction

We consider models for the spatio-temporal evolution of populations of microorganisms, moving in an incompressible fluid, which are able to partially orient their motion along gradients of a chemical signal. According to modeling approaches accounting for the mutual interaction of the swimming cells and the surrounding fluid, we study study parabolic chemotaxis systems coupled to the (Navier-)Stokes equations through transport and buoyancy-induced forces.
The presentation discusses mathematical challenges encountered even in the context of basic issues such as questions concerning global existence and boundedness, and attempts to illustrate this by reviewing some recent developments. A particular focus will be on strategies toward achieving a priori estimates which provide information sufficient not only for the construction of solutions, but also for some qualitative analysis.

Yu Jin
Population dynamics in varying river environments
To fully understand population dynamics in river ecosystems, it is necessary to recognize that rivers are subject to major seasonal variations in environmental factors that govern population growth and dispersal. In this talk, I will introduce some recent work about the studies of population spread and persistence in varying river environments by using integro-differential and integro-difference equations. We incorporate seasonal variations of population growth and dispersal into a time-periodic integro-differential equation model. Under the assumption of periodically fluctuating environments, we not only establish upstream and downstream spreading speeds to predict how fast the population spreads in the river but also address the critical domain size problem to determine how large a reach of suitable river habitat is needed to ensure population persistence of a stream-dwelling species. We also investigate the effect of seasonal correlations between the flow, transfer rates, diffusion and settling rates on the population spread and persistence. We then consider integro-difference models in the presence of advective flow with both periodic (alternating) and random kernel parameters. For the alternating kernel model, we obtain the principal eigenvalue of the linearization operator to determine population persistence. For the random model, we establish two persistence metrics: a generalized spectral radius and the asymptotic growth rate, which are mathematically equivalent but can be understood differently, to determine population persistence or extinction.

### Contributed Talks

University of Kansas
Tree harvesting in age-structured forests subject to beetle infestations
In this study we investigate a mathematical model for age-structured forest-beetle interactions. In the first part of this study, we consider different scenarios of the forest infestation by the beetle and observe that the quantitative age profile of the forest is significantly dependent on whether the beetle population is in endemic or epidemic state. In the second part we include harvesting of the forest trees and analyze two different harvesting strategies: clear cutting trees older than a certain age, and using a cut rate proportional to the number of trees older than a certain age. Numerical simulations are implemented to determine the optimal cutting age for both harvesting strategies. The numerical simulations reveal that, independent of the beetle population’s steady state (that is, no beetles, endemic or epidemic state) clear cutting all trees older than a given age provides a higher harvesting benefit. Our numerical simulations further indicate that in order to obtain a fixed harvesting yield, a forest under an beetle epidemic state have to be cut at a younger age than if the forest were at an endemic beetle state or a no beetle state.

Paulo Amorim
Federal University of Rio de Janeiro
Ant foraging dynamics: From reaction-diffusion to individual-based models.
I will present two approaches to modeling ant foraging behaviour. The first model is a system of parabolic PDEs of chemotaxis type and includes trail-laying and trail-following behaviour. I will present numerical simulations and also well-posedness results obtained with R. Alonso and T. Goudon. Finally, I will address some limitations of the model and present a new individual-based model producing accurate trail following behavior.

Amin Boumenir
University of West Georgia
Inverse problem for parabolic systems
We are concerned with the reconstruction of the coefficients of a linear parabolic system from finite time observations of the solution on the boundary. We present two procedures depending on whether the spectrum of the linear system is simple or multiple.

Kokum De Silva
Department of Mathematics, University of Peradeniya, Sri Lanka
Advection control in parabolic PDE systems for competitive populations
This paper investigates the reaction of two competing species to a given resource using optimal control techniques. We explore the choices of directed movement through controlling the advective coefficients in a system of parabolic partial differential equations. The objective is to maximize the abundance represented by a weighted combination of the two populations while minimizing the ’risk’ costs caused by the movements.

Luan Hoang
Texas Tech University
Regular solutions of the SKT system in any dimensions
We investigate the global time existence of smooth solutions for the Shigesada-Kawasaki-Teramoto (SKT) system of cross-diffusion equations of two competing species in population dynamics. If there are self-diffusion in one species and no cross-diffusion in the other, we show that the system has a unique smooth solution for all time in bounded domains of any dimension. We obtain this result by deriving global $$W^{1,p}$$-estimates of Calderón-Zygmund type for a class of nonlinear reaction-diffusion equations with self-diffusion. These estimates are achieved by employing Caffarelli-Peral perturbation technique together with a new two-parameter scaling argument.

Qihua Huang
University of Alberta
An invasion dynamics of zebra mussels in rivers and competitive interactions between zebra and quagga mussels.
While some species spread upstream in river environments, not all invasive species are successful in spreading upriver. Here the dynamics of unidirectional water flow found in rivers can play a role in determining invasion success. We develop a hybrid continuous/discrete-time model to describe the dynamics of invasive freshwater mussels in rivers. The model is used to understand how the interactions between population growth and dispersal, river flow rate, and water temperature, affect both persistence and the spread of zebra mussels along a river. We then investigate how the environmental factors, such as temperature and turbidity, affect competitive exclusion and coexistence of zebra and quagga mussels, based on a competition model.

King-Yeung Lam
The Ohio State University
Persistence Results of a PDE Population Model of Phytoplankton with Ratio Dependence
We study a PDE system modeling the growth of a single population consuming two forms of inorganic carbon in an unstirred chemostat. The resource uptake rate depends on the internal carbon storage and introduces a ratio dependence in the model. We first discuss the appropriate space for existence of solution that is motivated by the underlying biology. We will then proceed to apply a recent generalization of Krein-Rutman Theorem involving two different positive cones, due to Mallet-Paret and Nussbaum, to determine the threshold phenomena regarding persistence and extinction. This is joint work with Feng-Bin Wang (Chang Gung University, Taiwan) and Sze-Bi Hsu (National Tsing-Hua University, Taiwan).

Tien Khai Nguyen
PSU Mathematics Department
Conservation laws and some applications to travelingc ows
In this talk, I will introduce a new class of models of traffic flow on a network of roads. In these models models, the percentage of drivers who travel along an incoming road and wish to turn into an outgoing road is not a constant. Moreover, the drivers who enter a congested road are placed in a buffer of limited capacity, waiting their turn in line. The main goal is to describe traffic flow at intersections and study optimization problems on a network of roads. I will present the well-posedness result for a new intersection model of traffic flows, and the existence of globally optimal solutions, Nash equilibrium solutions for a decision problem involving a continuum of drivers on the network.

Samares Pal
University of Kalyani
Algae-herbivore interactions with Allee effect and chemical defense
Macroalgae exhibit a variety of characteristics that provide a degree of protection from herbivores. One characteristic is the production of chemicals that are toxic to herbivores. The toxic effect of macroalgae on herbivorous reef fish is studied by means of a spatiotemporal model of population dynamics with a nonmonotonic toxin-determined functional response of herbivores. The growth rate of macroalgae is mediated by Allee effect. We see that under certain conditions the system is uniformly persistent. Conditions for local stability of the system are obtained with weak and strong Allee effects. We observe that in presence of Allee effect on macroalgae, the system exhibits complex dynamics including Hopf bifurcation and saddle-node bifurcation.

### Poster Sessions

Stephen Colegate
Miami University, Oxford, OH
Mathematical modeling of drug use: the dynamics of monosubstance dependence for two addictive drugs.

Mustafa Elmas
University of Tennessee
A Model of Bacterial Chemotaxis and Its Simulation

Benjamin Levy
University of Tennessee
Modeling Feral Hogs in Great Smoky Mountains National Park

Buddhi Pantha
University of Tennessee
Modeling anthrax outbreak in wild animals and effects of control strategies.