#### Title

The Mathematical Modelling of Spatial Structure of Ecological System in Heterogeneous Environment

#### Publication Date

2017-05-09

#### Availability

Open access

#### Embargo Period

2017-05-09

#### Degree Type

Dissertation

#### Degree Name

Doctor of Philosophy (PHD)

#### Department

Biology (Arts and Sciences)

#### Date of Defense

2017-03-31

#### First Committee Member

Donald L. DeAngelis

#### Second Committee Member

Carol C. Horvitz

#### Third Committee Member

Leonel Sternberg

#### Fourth Committee Member

Chris Cosner

#### Fifth Committee Member

Daniel B. Botkin

#### Sixth Committee Member

Wei-Ming Ni

#### Abstract

Why do we need mathematical modelling Ecological modelling yields more general understanding and theory and provides testable and robust predictions. In particular, it is currently reaching the “next level” towards predictive and re-usable theory that can support environmental decision-making (Evans et al. 2013b). Therefore, in this dissertation work, I applied mathematical modelling to bridge pure mathematic theory with real ecology problems into two sections: (1) testing and understanding the impact of dispersal on total population size in a heterogeneous environment; (2) understanding and simulating the impact of biological control on an invasive plant and the long term dynamic change of the ecosystem in southern Florida. Could we have larger total population than total carrying capacity in a heterogeneous environment? Carrying capacity is a fundamental concept in ecology. An assumption in most non-spatial population models is that there is an upper limit on the size of the population, its carrying capacity, which is governed by the limiting resource. For example, for a plant population, this is typically space, light, or a nutrient. When the concept of carrying capacity is extended to an environment of spatially heterogeneous resources, the usual approach is to assume that the summation over the local carrying capacities yields the total carrying capacity of the whole domain. However, when the population disperses randomly in this domain, mathematical models predict that the upper limit on population size is no longer the summation over local carrying capacities. In studying a population in a two-patch system with logistic growth on each patch, where the per capita growth rates when the population is close to zero, r, and carrying capacities, K, differ on the two patches. When the two patches are connected by rapid diffusion and there is a relationship r1/K1 > r2/K2 for K1 > K2 between K and r of the two patches, the total population can reach a higher total steady state, or equilibrium, size than the sum of the subpopulations on the two patches without any connection. A mathematical derivation of a similar result was made, that considered a population of consumers in a continuous environment described by a reaction-diffusion equation with spatially varying carrying capacity (identical to the maximum growth rate), and showed that the total steady state size of a dispersed population exceeded the summation over all local carrying capacities for all diffusion rates. Further studies extended these results for both continuous spatial and multi-patch systems for populations with logistic growth in which parameters governing growth rate and carrying capacity could vary independently spatially, showing that the results held for small diffusion rates when a positive relationship existed between r and K, and for all diffusion rates when r is an accelerating convex function of K. Still, rigorous empirical validation of this "paradox" is generally lacking, so it is not known whether these results apply to real populations. Testing these results in the field or experimentally is further complicated by the fact that real populations are usually limited by exploitable resources, whereas the resources in previous models are assumed non-exploitable and not influenced by feedback from the consumer. Thus, it is not known how this more complex situation would change the results and other mathematical models. What is the long-term impact of biological control on an invasive species and our natural ecosystem? Melaleuca quinquenervia (Cav.) Blake (common names: melaleuca, paper bark, punk tree; Family, Myrtaceae, referred to as melaleuca thereafter) is a large (25-30m tall) native Australian tree introduced into the Florida landscape during the late 19th century for pulp production and ornamental purposes. It has strong invasive attributes, such as ecological fire adaptation and high reproductive potential. A single 10-m tall open-grown tree can store over 20 million seeds in its capsules at any given time. By the end of the 1900s melaleuca had spread over 200,000 ha of ecologically sensitive freshwater ecosystems of southern Florida displacing native vegetation such as slash pine (Pinus elliottii Engelm.) and pond cypress (Taxodium ascendens Brong.), threatening native biodiversity. Melaleuca invasion has caused adverse economic and environmental impacts to southern Florida, with the loss valued, 16 years ago, at nearly $30 million per year. Predicting the effects of invading species such as melaleuca is of current general interest because of the ecological and environmental damage of many invading species. The difficulty of making predictions of the establishment and spread has been pointed out. Modelling has been applied to make predictions of future spread in many cases, including both niche modeling and mechanistic models. Various control methods have been applied in many cases, including the use of biocontrol agents that are natural enemies of the pest species. Because use of both biocontrol and other methods of control is costly, prediction of the efficacy of control is equally urgent. The long-term success of biocontrol is still uncertain, so modeling has been used in a number of cases of invasive species, including plant species. Research objectives: The main objective of my dissertation research is to contribute to addressing these two questions as follows: In Chapter 2, I first aimed to determine if the mathematical result and others has relevance to empirical systems. That is, will a diffusing population in an environment with spatially varying resources reach a higher total equilibrium biomass than the population in the same environment without diffusion? The second objective is to test the mathematical result that a hump-shaped pattern appears when the equilibrium biomass is plotted as a function of the rate of diffusion. In Chapter 3, I tested three hypotheses suggested by the earlier mathematical results. Hypothesis 1: when a consumer exists in a domain with a heterogeneously distributed input of exploitable limiting resource, the steady state population can reach a greater size when it disperses than when it does not. Hypothesis 2: the higher population in a heterogeneous environment with diffusion is concomitant with a positive relationship of growth rate and carrying capacity. Hypothesis 3: a consumer population diffusing randomly in a domain with a heterogeneously distributed input of exploitable limiting resource can reach a greater steady state size than a population diffusing (or not) in a domain with the same total input of resources spread homogeneously in the domain. We utilized a budding yeast population to test these hypotheses experimentally, and, thereafter, used mathematical analysis to extend previous mathematical models to this case of exploitable resources. In Chapter 4, the objective is to improve understanding of the possible effects of herbivory on the landscape dynamics of melaleuca in native southern Florida plant communities. To do that, I projected likely future changes in plant communities using the individual based modeling platform, JABOWA-II, by simulating successional processes occurring in two types of southern Florida habitat, cypress swamp and bay swamp, occupied by native species and melaleuca, with the impact of insect herbivores. In Chapter 5, my goal is to estimate the rate of defoliation needed to achieve a specified reduction in the growth rate under various conditions of nutrient availability to the tree and how it might change its allocations to foliage and roots in an optimal way.

#### Keywords

Heterogeneity, Mathematical Modelling, Ecology

#### Recommended Citation

Zhang, Bo, "The Mathematical Modelling of Spatial Structure of Ecological System in Heterogeneous Environment" (2017). *Open Access Dissertations*. 1865.

https://scholarlyrepository.miami.edu/oa_dissertations/1865