MS07 - ECOP-02

Advances in Spatial Ecological and Epidemiological Modeling and Analysis (Part 1)

Thursday, July 17 at 4:00pm

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Note: this minisymposia has multiple sessions. The other session is: and Part-2.

Daozhou Gao (Cleveland State University), Xingfu Zou, University of Western Ontario

Description:

Animal dispersal and human movement play a key role in ecology and epidemiology. Species with limited dispersal abilities are more likely to go extinct, especially in the face of habitat fragmentation and climate change. Meanwhile, massive travel and tourism accelerate the global spread of infectious agents. Mathematical models serve as powerful tools for describing and predicting population growth and the spatial-temporal spread of diseases. There are strong connections between population dynamics and disease dynamics, particularly when spatial heterogeneity and population mobility are concerned. By integrating ecological and epidemiological insights, such models provide valuable frameworks for conservation strategies, public health interventions, and policy planning. Despite substantial advances in model formulations, analysis and applications, some major challenges persist. These challenges include accounting for nonlinear interactions, understanding pathogen persistence in fluctuating environments, and addressing computational limitations in high-dimensional systems. The eight invited talks will cover topics on (1) diffusive population model; (2) pathogen persistence in variable environments; (3) partially degenerate reaction-diffusion system; (4) toxicant-taxis model. This mini-symposium provides a great opportunity to showcase some recent progresses in addressing these challenges, exchange ideas and foster interdisciplinary collaborations among diverse researchers.

Sebastian Schreiber

University of California, Davis

"Impacts of the Tempo and Mode of Environmental Fluctuations on Population Growth"

Populations consist of individuals living in different states and experiencing temporally varying environmental conditions. Individuals may differ in their geographic location, stage of development (e.g. juvenile versus adult), or physiological state (infected or susceptible). Environmental conditions may vary due to abiotic (e.g. temperature) or biotic (e.g. resource availability) factors. As survival, growth, and reproduction of individuals depend on their state and the environmental conditions, environmental fluctuations often impact population growth. Here, we examine to what extent the tempo and mode of these fluctuations matter for population growth. We model population growth for a population with $d$ individual states and experiencing $N$ different environmental states. The models are switching, linear ordinary differential equations $x'(t)=A(sigma(omega t))x(t)$ where $x(t)=(x_1(t),dots,x_d(t))$ corresponds to the population densities in the $d$ individual states, $sigma(t)$ is a piece-wise constant function representing the fluctuations in the environmental states $1,dots,N$, $omega$ is the frequency of the environmental fluctuations, and $A(1),dots,A(n)$ are Metzler matrices representing the population dynamics in the environmental states $1,dots,N$. $sigma(t)$ can either be a periodic function or correspond to a continuous-time Markov chain. Under suitable conditions, there exists a Lyapunov exponent $Lambda(omega)$ such that $lim_{ttoinfty} frac{1}{t}logsum_i x_i(t)=Lambda(omega)$ for all non-negative, non-zero initial conditions $x(0)$ (with probability one in the random case). For both random and periodic switching, we derive analytical first-order and second-order approximations of $Lambda(omega)$ in the limits of slow ($omegato 0$) and fast ($omegatoinfty$) environmental fluctuations. When the order of switching and the average switching times are equal, we show that the first-order approximations of $Lambda(omega)$ are equivalent in the slow-switching limit, but not in the fast-switching limit. Hence, the mode (random versus periodic) of switching matters for population growth. We illustrate our results with applications to a simple stage-structured model and a general spatially structured model. When dispersal rates are symmetric, the first order approximations suggest that population growth rates increase with the frequency of switching -- consistent with earlier work on periodic switching. In the absence of dispersal symmetry, we demonstrate that $Lambda(omega)$ can be non-monotonic in $omega$. In conclusion, our results show that population growth rates often depend both on the tempo ($omega$) and mode (random versus deterministic) of the environmental fluctuations. This work is in collaboration with Pierre Monmarch'{e} (Institut universitaire de France) and '{E}douard Strickler (Universit'{e} de Lorraine).

Adrian Lam

Ohio State University

"Can Spatial Heterogeneity Alone Lead to Selection for Dispersal?"

In a seminal paper, A. Hastings showed by pairwise invasbility analysis that in a stationary environment, species modeled by the diffusive logistic equation evolves towards smaller diffusion rate. A particular consequence is that in the competition model of N species differing only by the diffusion rate, an equilibrium is locally stable if and only if it is the one dominated by the slowest moving species, assuming other factors are equal. It is conjectured that such an equilibrium is also globally attractive, namely, the slowest moving species always competitively exclude all other species. In the first part of the talk, we survey some recent progress on the conjecture. In the second part of the talk, we describe a recent competition experiment with nematode worm population performed by B. Zhang, in which fast dispersal can be advantageous even though there is little temporal fluctuation in the environment. Motivated by the experimental observation, we introduce a single population model where the organism switches between two alternative physiological states (high vs low food storage), and demonstrate that for such a population model, fast diffusion can be selected in stationary environments.

Yijun Lou

The Hong Kong Polytechnic University

"Dynamics of a Reaction-diffusion System with Time-periodic and Spatially Dependent Delay: A Quotient Space Approach"

Understanding the impact of temporal and spatial heterogeneity on population dynamics has significantly advanced mathematical theories in reaction-diffusion systems. This talk reports the global dynamics of a reaction-diffusion system with time delay, where all model parameters are spatially and temporally dependent. The main focus lie in threefold: (i) formulating a stage-structured model that incorporates diffusion and temporally and spatially inhomogeneous development delay; (ii) proposing a dynamical systems framework that employs a quotient phase space to establish strong monotonicity of the periodic semiflow; (iii) establishing a threshold-type result under minimal assumptions, for both increasing and unimodal birth functions. This synthesized approach is expected to motivate further studies when strong monotonicity of the solution semiflow fails in conventional phase spaces.

Chunyi Gai

University of Northern British Columbia

"Resource-mediated Competition between Two Plant Species with Different Rates of Water Intake"

We propose an extension of the well-known Klausmeier model of vegetation to two plant species that consume water at different rates. Rather than competing directly, the plants compete through their intake of water, which is a shared resource between them. In semi-arid regions, the Klausmeier model produces vegetation spot patterns. We are interested in how the competition for water affects the co-existence and stability of patches of different plant species. We consider two plant types: a “thirsty” species and a “frugal” species, that only differ by the amount of water they consume per unit growth, while being identical in other aspects. We find that there is a finite range of precipitation rate for which two species can co-exist. Outside of that range (when the rate is either sufficiently low or high), the frugal species outcompetes the thirsty species. As the precipitation rate is decreased, there is a sequence of stability thresholds such that thirsty plant patches are the first to die off, while the frugal spots remain resilient for longer. The pattern consisting of only frugal spots is the most resilient. The next-most-resilient pattern consists of all-thirsty patches, with the mixed pattern being less resilient than either of the homogeneous patterns. We also examine numerically what happens for very large precipitation rates. We find that for a sufficiently high rate, the frugal plant takes over the entire range, outcompeting the thirsty plant.

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Annual Meeting for the Society for Mathematical Biology, 2025.