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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. 

Blue mussels (Mytilus edulis) are important keystone species that have been declining in the Gulf of Maine. This could be attributed to a variety of complex factors such as indirect effects due to invasion by epibionts, which remains unexplored mathematically. Based on classical optimal foraging theory (OFT) and anti-fouling defense mechanisms of mussels, we derive an ODE model for crab–mussel interactions in the presence of an invasive epibiont, Didemnum vexillum. The dynamical analysis leads to results on stability, global boundedness and bifurcations of the model. Next, via optimal control methods, we predict various ecological outcomes. Our results have key implications for preserving mussel populations in the advent of invasion by non-native epibionts. In particular, they help us understand the changing popluation dynamics of local predator–prey communities, due to indirect effects that epibionts confer. 

Abstract The soybean aphid, Aphis glycines (Hemiptera: Aphididae), is an invasive pest that can cause severe yield loss to soybeans in the North Central United States. A tactic to counter this pest is the use of aphid-resistant soybean varieties. However, the frequency of virulent biotypes that can survive on resistant varieties is expected to increase as more farmers use these varieties. Soybean aphids can alter soybean physiology primarily by two mechanisms, feeding facilitation, and the obviation of resistance, favoring subsequent colonization by additional conspecifics. We developed a nonlocal, differential equation population model to explore the dynamics of these biological mechanisms on soybean plants coinfested with virulent and avirulent aphids. We then use demographic parameters from laboratory experiments to perform numerical simulations via the model. We used this model to determine that initial conditions are an important factor in the season-long cooccurrence of both biotypes. The initial population of both biotypes above the resistance threshold or avirulent aphid close to resistance threshold and high virulent aphid population results in coexistence of the aphids throughout the season. These simulations successfully mimicked aphid dynamics observed in the field- and laboratory-based microcosms. The model showed an increase in colonization of virulent aphids increases the likelihood that aphid resistance is suppressed, subsequently increasing the survival of avirulent aphids. This interaction produced an indirect, positive interaction between the biotypes. These results suggest the potential for a ‘within plant’ refuge that could contribute to the sustainable use of aphid-resistant soybeans.