Some mathematical problems in population ecology

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Takyi, Eric Mawuena
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Parshad, Rana
Rossmanith, James
Kadelka, Claus
Wu, Zhijun
Rogers, Haldre
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We study an investigated biological control method of eradicating invasive species. This method is known as the Trojan Y Chromosome (TYC) strategy. It is applicable to species with XX-XY determinism systems wherein a genetically modified organism is introduced into the invasive population to skew the sex ratio towards males over generations thereby causing extinction. In this thesis, we study the classical three species TYC model and show that, introducing supermales above certain thresholds can lead to negative solutions for the male population which in turn can lead to finite time blow-up in the female and male population. This is biologically unrealistic. Therefore, we investigate improvements to the TYC modeling construct and propose four new models. Each of these new models is dynamically explored and is shown to possess globally existing solutions in any parameter and data regime. We also study an alternative control strategy which mimics the TYC strategy called the female harvesting male stocking (FHMS). We consider the FHMS strategy with a weak Allee effect and show that, the extinction boundary need \textit{not} be hyperbolic. We give the first example of a non-hyperbolic extinction boundary in mating systems structured by sex to the best of our knowledge. We investigate certain two species ODE and PDE Lotka-Volterra competition models where one of the competitors has a likelihood of going extinct in finite time. We see that, the weaker competitor can escape competitive exclusion with finite time extinction mechanism present and the slower diffusing competitor may not always win. We support our results with numerical simulations. Finally we show finite time blow-up solutions in three species modified Leslie-Gower system with mutual interference and prey defense for sufficiently large or small initial data. We also show that a tritrophic food chain model subject to a Allee effect on the prey growth and Crowley-Martin senses functional response between intermediate predator and top predator, with a top predator of sexually reproductive type blows-up under certain parametric restrictions on the parameter space. We derive a new extinction boundary for the system.We also conjecture on the effect of the Allee threshold on the blow-up dynamics in the model. All of our results are validated via numerical simulations.
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