國立虎尾科技大學 |

Traveling Waves in Cyclic Ecological Systems.

Record Type:	Language materials, manuscript : Monograph/item
Title/Author:	Traveling Waves in Cyclic Ecological Systems./
Author:	Wang, Zihao.
Description:	1 online resource (195 pages)
Notes:	Source: Dissertations Abstracts International, Volume: 85-06, Section: B.
Contained By:	Dissertations Abstracts International85-06B.
Subject:	Applied mathematics. -
Online resource:	click for full text (PQDT)
ISBN:	9798381175431

Traveling Waves in Cyclic Ecological Systems.
Wang, Zihao.

Traveling Waves in Cyclic Ecological Systems. - 1 online resource (195 pages)

Source: Dissertations Abstracts International, Volume: 85-06, Section: B.

Thesis (Ph.D.)--Northwestern University, 2023.

Includes bibliographical references

This dissertation is concerned with mathematical models for how different species (of animals, plants, bacteria, or other diffusion things) spread out and compete with one another. It examines the dynamics of traveling wave solutions within multi-species cyclic ecological systems. We develop, analyze and simulate deterministic models mainly for two and four-species ecosystems, which are described by systems of reaction-diffusion partial differential equations. These models account for species mobility via Fickian diffusion and inter-species interactions based on a cyclic competition scheme. The study incorporates analytical methods, specifically asymptotic and perturbative approaches, alongside numerical simulations, to investigate in detail the inter-species competition dynamics, traveling wave propagation phenomena, and the determination of dominant species or equilibrium states. In the context of species invasions, we show that direct inter-species competition, mobility, and birthrates, as well as their interrelationships with one another, are the factors that determine the dominant species and long-term dynamics of cyclic systems. These key factors are expressed by specific parameters within the deterministic systems, and asymptotic solutions are presented across various limiting parameter regimes, which are further substantiated by numerical simulations.We initially examine traveling wave solutions of the Lotka-Volterra competition-diffusion system for two species in the bistable region. This includes an investigation into the propagation speed of traveling waves and, most importantly, the sign of the speed, which determines the winner of the competition, in several limiting cases. These cases are categorized into (i) strong competition, where inter-species competition for each species notably surpasses intra-species competition, and (ii) moderate competition, where inter-and intra-species competitions are of similar strength. Furthermore, scenarios involving fast and slow competitors-defined by significantly different diffusion coefficients-are analyzed.Building upon the Lotka-Volterra two-species competition model, we study a four-species cyclic community. Our research reveals intriguing dynamics in the four-species cyclic ecosystem. We observe the formation of two stable alliances between non-competing species in the parameter regimes where inter-species competition exceeds intra-species competition. We derive conditions under which one alliance can displace the other in an " invasion " scenario. In particular, we examine the situation where both alliances are evenly matched - a state referred to as the " standstill problem. " Our investigation focuses on three specific parameter regimes: (i) moderate competition, (ii) strong competition, and (iii) slow competitors (diffusivities for one alliance are asymptotically close to zero, or, alternatively, birthrates for one alliance are asymptotically large). Analytical results obtained from this section are cross-verified with numerical computations and previous findings related to the two-species model, demonstrating a high degree of agreement. Lastly, we collect the accumulated research findings and introduce two new methodologies: piecewise linear approximation and front propagation analysis. The former method, which involves linearizing the system in different spatial regions and subsequently matching to estimate the propagation speed, encountered challenges. Specifically, the piecewise linear method was not entirely effective for the traveling wave problems. The differences between the approximate solutions and the numerical simulations were likely attributable to the complexity of the interspecies interface. Direct derivative matching in this region seems to neglect many vital details, leading to a solution that lacks sufficient accuracy. However, this method was not without value. Despite its limitations, it provided instructive insights that inspired us to reconsider our approach. Drawing from these preliminary findings, we developed alternative methodologies, as detailed in Chapters 2 and 3. On the other hand, the front propagation analysis method offers an estimation technique for propagation speeds of unstable states in the presence of pulled fronts. Propagation speeds are computed and estimated in several asymptotic limit cases. For four-species systems, the interface between the advancing and retreating states is intricate, and intermediate buffer states can emerge under specific parameter regimes. When this happens, pulled fronts appear, and this method proves effective in estimating the propagation speed between states.

Electronic reproduction.
Ann Arbor, Mich. :
ProQuest,
2024

Mode of access: World Wide Web

ISBN: 9798381175431Subjects--Topical Terms:

1069907
Applied mathematics.
Subjects--Index Terms:

Front propagation analysis Index Terms--Genre/Form:

554714
Electronic books.

Traveling Waves in Cyclic Ecological Systems.
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500 $a Advisor: Volpert, Vladimir;Bayliss, Alvin.
502 $a Thesis (Ph.D.)--Northwestern University, 2023.
504 $a Includes bibliographical references
520 $a This dissertation is concerned with mathematical models for how different species (of animals, plants, bacteria, or other diffusion things) spread out and compete with one another. It examines the dynamics of traveling wave solutions within multi-species cyclic ecological systems. We develop, analyze and simulate deterministic models mainly for two and four-species ecosystems, which are described by systems of reaction-diffusion partial differential equations. These models account for species mobility via Fickian diffusion and inter-species interactions based on a cyclic competition scheme. The study incorporates analytical methods, specifically asymptotic and perturbative approaches, alongside numerical simulations, to investigate in detail the inter-species competition dynamics, traveling wave propagation phenomena, and the determination of dominant species or equilibrium states. In the context of species invasions, we show that direct inter-species competition, mobility, and birthrates, as well as their interrelationships with one another, are the factors that determine the dominant species and long-term dynamics of cyclic systems. These key factors are expressed by specific parameters within the deterministic systems, and asymptotic solutions are presented across various limiting parameter regimes, which are further substantiated by numerical simulations.We initially examine traveling wave solutions of the Lotka-Volterra competition-diffusion system for two species in the bistable region. This includes an investigation into the propagation speed of traveling waves and, most importantly, the sign of the speed, which determines the winner of the competition, in several limiting cases. These cases are categorized into (i) strong competition, where inter-species competition for each species notably surpasses intra-species competition, and (ii) moderate competition, where inter-and intra-species competitions are of similar strength. Furthermore, scenarios involving fast and slow competitors-defined by significantly different diffusion coefficients-are analyzed.Building upon the Lotka-Volterra two-species competition model, we study a four-species cyclic community. Our research reveals intriguing dynamics in the four-species cyclic ecosystem. We observe the formation of two stable alliances between non-competing species in the parameter regimes where inter-species competition exceeds intra-species competition. We derive conditions under which one alliance can displace the other in an " invasion " scenario. In particular, we examine the situation where both alliances are evenly matched - a state referred to as the " standstill problem. " Our investigation focuses on three specific parameter regimes: (i) moderate competition, (ii) strong competition, and (iii) slow competitors (diffusivities for one alliance are asymptotically close to zero, or, alternatively, birthrates for one alliance are asymptotically large). Analytical results obtained from this section are cross-verified with numerical computations and previous findings related to the two-species model, demonstrating a high degree of agreement. Lastly, we collect the accumulated research findings and introduce two new methodologies: piecewise linear approximation and front propagation analysis. The former method, which involves linearizing the system in different spatial regions and subsequently matching to estimate the propagation speed, encountered challenges. Specifically, the piecewise linear method was not entirely effective for the traveling wave problems. The differences between the approximate solutions and the numerical simulations were likely attributable to the complexity of the interspecies interface. Direct derivative matching in this region seems to neglect many vital details, leading to a solution that lacks sufficient accuracy. However, this method was not without value. Despite its limitations, it provided instructive insights that inspired us to reconsider our approach. Drawing from these preliminary findings, we developed alternative methodologies, as detailed in Chapters 2 and 3. On the other hand, the front propagation analysis method offers an estimation technique for propagation speeds of unstable states in the presence of pulled fronts. Propagation speeds are computed and estimated in several asymptotic limit cases. For four-species systems, the interface between the advancing and retreating states is intricate, and intermediate buffer states can emerge under specific parameter regimes. When this happens, pulled fronts appear, and this method proves effective in estimating the propagation speed between states.
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538 $a Mode of access: World Wide Web
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650 4 $a Ecology. $3 575279
650 4 $a Systematic biology. $3 1179695
653 $a Front propagation analysis
653 $a Inter-species interactions
653 $a Instructive insights
653 $a Propagation speeds
653 $a Four-species systems
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690 $a 0423
690 $a 0329
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710 2 $a Northwestern University. $b Engineering Sciences and Applied Mathematics. $3 1180565
773 0 $t Dissertations Abstracts International $g 85-06B.
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