Abstract
The Trojan Y Chromosome Strategy (TYC) is the only genetic biological control method in practice in North America for controlling invasive populations with an XX–XY sex determinism. Herein a modified organism, that is a supermale or feminised supermale, is introduced into an invasive population to skew the sex ratio over time, causing local extinction. We consider the three species TYC reaction diffusion model, and show that introduction of supermales above certain thresholds, and for certain initial data, solutions can blow-up in finite time. Thus, in order to have biologically meaningful solutions, one needs to restrict parameter and initial data regimes, in TYC type models.
Highlights
Biological control is an alternative means of control, where a species is released to control the invasive population by predation, competition, disease, or manipulating the mating system [9,10]
The previous theorem proves that the finite time blow-up is a possibility even if the only introduction of s is through the initial condition
In the following theorem we prove that regardless of the initial condition size that there exists a threshold to the introduction rate, μ, such that rates beyond this value will lead to finite time blow-up in f
Summary
The detrimental effects of invasive species is well-documented [1–7]. Current control methods rely, primarily, on chemical treatment [8] and are environmentally detrimental. Prior work on TYC [25–28], shows that depending on the introduction rate of the supermale/feminized supermale, and initial data, eradication is always possible These results rely on the assumption that solutions remain positive for all non-negative initial data. What we have in practice is a situation where only a YY supermale is introduced into a population of XX females and XY males This leads to the following reaction diffusion system:. The three species reaction diffusion TYC model, (1)–(3), results in negative solutions in the male population m. These conditions include the choice of large initial conditions ( f0 + m0 + s0 >> K) and/or large introduction rate of supermales s.
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