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Item Understanding multiple species ecosystem dynamics using a consumer resource model(Wiley, 2016) Collins, Obiora Cornelius; Duffy, Kevin JanMost ecological systems comprise multiple species coex-isting and the dynamics of these multiple species can be important for understanding, management, and conservation. One method to study such ecological system dynamics is the use of heterogeneous models. Here we for-mulate and analyze a multiple species (n patches or groups) consumer re-source model. Initial insights are gained by analyzing the special cases n =1 and n = 2. A threshold consumption number C0 is used to investigate system stability and hence the long-term dynamics of the system. It is shown how this threshold consumption number can measure the effects and extent of multiple species coexistence in the system.Item Consumption threshold used to investigate stability and ecological dominance in consumer-resource dynamics(Elsevier, 2015) Collins, Obiora Cornelius; Duffy, Kevin JanUnderstanding consumer resource population dynamics can be important to an understanding of the overall ecology of systems. For example, the tree-grass continuum dynamics of savannas, an important ecological biome, is influenced by the population dynamics. Here we investigate herbivory driven popu-lation dynamics of a savanna using a simple model of the interactions of the dominant players, namely: trees, grasses, browsers, grazers and mixed browsers-grazers. We introduce a consumption threshold that summarises some of the parameters and this is used as a guide to understanding the dynamics. This number is used in investigating system stability and sensitivity to parameter fluctuations. It is also used to identify degrees of ecological dominance.Item Optimal control of maize foliar diseases using the plants population dynamics(Taylor and Fancis, 2015-09-28) Collins, Obiora Cornelius; Duffy, Kevin JanPathogens and insects can have important negative effects on yields of crops cultivated by humans. These effects can be important for the food security or financial well-being of individuals. In particular, maize is a very important staple crop worldwide and is vulnerable to diseases. We formulate here a mathematical model to evaluate the impacts of foliar diseases on the population dynamics of maize plants. Qualitative analyses of the important mathematical features of the model are carried out. We show how this methodology can be extended to reducing the spread of foliar diseases through effective control measures with minimum costs.Item Identifying stability conditions and Hopf bifurcations in a consumer resource model using a consumption threshold(Elsevier, 2016) Duffy, Kevin Jan; Collins, Obiora CorneliusThe existence, or not, of cyclic dynamics is one of the pivotal aspects of ecological populations. This work considers a consumer resource model found in ecology that can describe both cyclic and non-cyclic dynamics depending on parameter conditions. A threshold consumption number C0 is introduced, similar to the basic reproduction in epidemiological models. It is shown that consumer survival requires C0 > 1 and that a Hopf bifurcation occurs at , where is defined here and is greater than 1. This result is discussed with an example and extensions to other more complicated models.