| Author: | Sun, Bei |
| Title: | Population dynamics of frog species under seasonal factors and disease transmission |
| Advisors: | Lou, Yijun (AMA) |
| Degree: | Ph.D. |
| Year: | 2025 |
| Department: | Department of Applied Mathematics |
| Pages: | xv, 203 pages : color illustrations |
| Language: | English |
| Abstract: | Frog populations are essential components of the ecosystem, and their decline or even extinction will significantly harm biodiversity and undermine the stability of the ecosystem. The growth and reproduction of frog population are essentially determined by the presence of seasonal climate conditions and ecological environments suitable for their survival. Comprehending the dynamics of frog populations in response to seasonal weather variations is crucial for forecasting population trends and devising conservation strategies for amphibians in the context of climate change scenarios. The influence of individual behaviors, such as frog mating, on population dynamics is also a worthy subject of exploration. In addition, worldwide amphibian decline and extinction have been observed, highlighting the importance of identifying the underlying factors. This issue has long been recognized as highly significant and continues to receive substantial attention in conservation ecology. Pathogen infection, in particular the chytrid fungus Batrachochytrium dendrobatidis, is postulated as a key factor contributing to the decline of certain species within specific regions. These issues will be investigated in this thesis using stage-structured models, which group individuals with similar demographic characteristics together and have proven useful in describing population dynamics. This thesis begins with a brief introduction in Chapter 1 grounded in both biological perspectives and mathematical motivations. This chapter explains the ecological and mathematical significance of exploring the population dynamics of frog species. In Chapter 2 we present a concise overview of the mathematical foundations, elucidating key mathematical terminologies and theorems integral to monotone dynamical systems, global attractors, uniform persistence, coexistence states, and the basic reproduction ratio within the context of population models in periodic environments. Chapter 3 starts from reviewing two widely-used modeling frameworks, in the form of integral equations and age-structured partial differential equations. Both modeling frameworks can be reduced into same differential equation structures with/without time delays under Dirac and gamma distributions for the stage durations. Each framework has its advantages and inherent limitations. The net reproduction number and initial growth rate can be easily defined from the integral equation. However, it becomes challenging to integrate the density-dependent regulations on the stage distribution and survival probabilities in an integral equation, which may be suitably incorporated in partial differential equations. In Chapter 4, we formulate a stage-structured frog population model in the ecological environment with temperature-dependent effects. Due to the consideration of seasonal developmental duration, the resulting model is a system of piecewise differential equations that incorporate temperature-dependent delays. We propose the quotient space based on the initial natural phase space and prove the strong monotonicity in addition to showing some basic properties of the solutions. We demonstrate that the basic reproduction number, R0, serves as a critical threshold parameter that dictates whether the frog population will go extinct or persist. According to the theory of monotone dynamical systems, asymptotically periodic semiflows, and the comparison method, we obtain the global dynamics of the frog population system. The final simulations verify the analytic results numerically. To focus on the pathogen characteristics that can drive host species extinction, both deterministic and stochastic modeling frameworks based on a susceptible-infectious-bacteria epidemic model are proposed in Chapter 5 to assess the influence of pathogen infection on species decline and extinction. Various indices, including the reproduction numbers of the host species, the replication of the pathogen, and the transmission of the pathogen are derived. Theoretical analysis includes the stability of equilibria, the extinction and persistence of host species in the deterministic model, and the evaluation of extinction probability and average extinction time in the stochastic model. Additionally, numerical simulations are conducted to quantify the effects of various factors on host decline and extinction, as well as the probabilities of extinction. We find two crucial conditions for a pathogen to drive host extinction: (i) the pathogen's self-reproduction capacity in the environment, and (ii) the pathogen's impact on the fecundity and survival of the infected host. These findings provide insights that could aid in the design and implementation of effective conservation strategies for amphibians. Chapter 6 develops a stage-structured model with periodic time-delay for frog populations, comprehensively incorporating factors such as seasonal succession, two-sex division, mating behaviors, and adult competition. This periodic succession model describes the dynamic characteristics of female and male frog populations during both the normal and hibernation periods. Based on this framework, we analyze the basic properties within the natural phase space, including existence and uniqueness, boundedness, monotonicity, and strict subhomogeneity. To further investigate strong monotonicity, we introduce the quotient space, employing a method similar to the analytical approach used in Chapter 4. The global dynamics of the population model are then obtained through the introduction of the basic reproduction number, the use of auxiliary systems, and a series of theories on monotone dynamical systems, periodic semiflows, and the comparison method. Numerical simulations illustrate the influence of time-dependent parameters and validate the related analytic results. Additionally, they assess the impact of two key sensitive parameters--mature mortality rates and mating pairs--on population sizes during the normal growth and hibernation periods across multiple life-cycles and a single life-cycle. The simulations clearly demonstrate that female and male populations ultimately experience significant declines as they approach hibernation. However, a higher number of mating pairs leads to a higher stabilized population size during the normal period before the decline associated with hibernation. Chapter 7 provides a summary of the results presented in this thesis and discusses potential directions for future research. |
| Rights: | All rights reserved |
| Access: | open access |
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