MS09 - MFBM-10

Flow-Kick Dynamics in Population Biology: Bridging Continuous and Discrete Processes (Part 2)

Friday, July 18 at 4:00pm

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Note: this minisymposia has multiple sessions. The other session is: and Part-1.

Sebastian Schreiber (University of California, Davis)

Description:

This minisymposium will explore recent advances in flow-kick (aka hybrid or impulsive) dynamical systems and their applications to population biology. Flow-kick models, which combine continuous dynamics with discrete perturbations of states and parameters, provide a powerful framework for studying biological systems that experience both smooth and abrupt changes. Recent theoretical developments have established new tools for analyzing the interplay between continuous and discrete processes in biological systems. Applications including consumer-resource dynamics with seasonal reproduction, epidemiological models with pulsed vaccination, and population management with periodic harvesting demonstrate how flow-kick approaches can capture emergent phenomena missed by purely continuous or discrete models. The minisymposium will bring together researchers developing mathematical theory and numerical methods for flow-kick systems alongside those applying these tools to concrete biological problems. Talks will showcase both analytical approaches and empirical applications, with emphasis on mechanistic understanding of population responses to perturbation. By highlighting this growing synthesis of theory and application, the session aims to stimulate new collaborations advancing our ability to understand and manage populations experiencing recurrent disturbances.

Junping Shi

College of William and Mary

"Effect of rotational grazing on plant and animal production"

It is a common understanding that rotational cattle grazing provides a better yield than continuous grazing, but a qualitative analysis is lacking in the agriculture literature. In rotational grazing, cattle periodically move from one paddock to another in contrast to continuous grazing, in which the cattle graze on a single plot for the entire grazing season. Here we quantitatively show how production yields and stockpiled forage are greater in rotational grazing in some harvesting models. We construct a vegetation grazing model on a fixed area, and by using parameters obtained from agricultural publications and keeping the minimum value of remaining forage constant, our result shows that both the number of cattle per acre and stockpiled forage increase for all tested rotational configurations than the continuous grazing. Some related spatial harvesting models are also discussed. This is a joint work with Mayee Chen.

Kate Meyers

Carleton College

"From deluges to drizzle: continuous limits of flow-kick models"

To incorporate ongoing disturbances into a differential equation (DE) model of biological processes, one might embed the disturbance continuously in the DE or resolve the disturbance discretely. In this talk we’ll explore the flow-kick approach to modeling repeated, discrete disturbances and examine the dynamic implications of this modeling choice. We’ll position continuous disturbances as limits of repeated, discrete ones and share recent results on how flow-kick systems both mimic and depart from their continuous analogs.

Rebecca Tyson

University of British Columbia, Okanagan

"Host-parasitoid systems are vulnerable to extinction via P-tipping: Forest Tent Caterpillar as an example"

Continuous-time predator-prey models admit limit cycle solutions that are vulnerable to the phenomenon of phase-sensitive tipping (P-tipping): The predator-prey system can tip to extinction following a rapid change in a key model parameter, even if the limit cycle remains a stable attractor. In this paper, we investigate the existence of P-tipping in an analogous discrete-time system: a host-parasitoid system, using the economically damaging forest tent caterpillar as our motivating example. We take the intrinsic growth rate of the consumer as our key parameter, allowing it to vary with environmental conditions in ways consistent with the predictions of global warming. We find that the discrete-time system does admit P-tipping, and that the discrete-time P-tipping phenomenon shares characteristics with the continuous-time one: Both require an Allee effect on the resource population, occur in small subsets of the phase plane, and exhibit stochastic resonance as a function of the autocorrelation in the environmental variability. In contrast, the discrete-time P-tipping phenomenon occurs when the environmental conditions switch from low to high productivity, can occur even if the magnitude of the switch is relatively small, and can occur from multiple disjoint regions in the phase plane. This is joint work with Bryce F. Dyck.

Sebastian Schreiber

University of California, Davis

"Coexistence and extinction in flow kick systems via Lyapunov exponents"

Natural populations experience a complex interplay of continuous and discrete processes: continuous growth and interactions are punctuated by discrete reproduction events, dispersal, and external disturbances. These dynamics can be modeled by impulsive or flow-kick systems, where continuous flows alternate with instantaneous discrete changes. To study species persistence in these systems, an invasion growth rate theory is developed for flow-kick models with state-dependent timing of kicks and auxiliary variables that can represent stage structure, trait evolution, and environmental forcing. The invasion growth rates correspond to Lyapunov exponents that characterize the average per-capita growth of species when rare. Two theorems are proven that use invasion growth rates to characterize permanence, a form of robust coexistence where populations remain bounded away from extinction. The first theorem uses Morse decompositions of the extinction set and requires that there exists a species with a positive invasion growth rate for every invariant measure supported on a component of the Morse decomposition. The second theorem uses invasion growth rates to define invasion graphs whose vertices correspond to communities and directed edges to potential invasions. Provided the invasion graph is acyclic, permanence and extinction are fully characterized by the signs of the invasion growth rates. Invasion growth rates are also used to identify the existence of extinction-bound trajectories and attractors that lie on the extinction set. To demonstrate the framework's utility, these results are applied to a microbial serial transfer model where state-dependent timing enables coexistence through a storage effect, a spatially structured consumer-resource model showing intermediate reproductive delays can maximize persistence, and an empirically parameterized Lotka-Volterra model demonstrating how disturbance can lead to extinction by disrupting facilitation. Mathematical challenges, particularly for systems with cyclic invasion graphs, and promising biological applications are discussed. These results reveal how the interplay between continuous and discrete dynamics creates ecological outcomes not found in purely continuous or discrete systems, providing a foundation for predicting population persistence and species coexistence in natural communities subject to gradual and sudden changes

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Annual Meeting for the Society for Mathematical Biology, 2025.