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Biological control via “ecological” damping: an approach that attenuates non-target effects. (English) Zbl 1364.92061

Summary: In this work we develop and analyze a mathematical model of biological control to prevent or attenuate the explosive increase of an invasive species population, that functions as a top predator, in a three-species food chain. We allow for finite time blow-up in the model as a mathematical construct to mimic the explosive increase in population, enabling the species to reach “disastrous”, and uncontrollable population levels, in a finite time. We next improve the mathematical model and incorporate controls that are shown to drive down the invasive population growth and, in certain cases, eliminate blow-up. Hence, the population does not reach an uncontrollable level. The controls avoid chemical treatments and/or natural enemy introduction, thus eliminating various non-target effects associated with such classical methods. We refer to these new controls as “ecological damping”, as their inclusion dampens the invasive species population growth. Further, we improve prior results on the regularity and Turing instability of the three-species model that were derived in [the first author et al., ibid. 254, 83–102 (2014; Zbl 1323.92182)]. Lastly, we confirm the existence of spatiotemporal chaos.

MSC:

92D40 Ecology
35K57 Reaction-diffusion equations
35B36 Pattern formations in context of PDEs
35B35 Stability in context of PDEs
92D25 Population dynamics (general)

Citations:

Zbl 1323.92182
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References:

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