Abstract. Cannibalism, which is the act of killing and at least partial consumption of conspecifics, is ubiquitous in nature. Mathematical models have considered cannibalism in the predator primarily, and show that predator cannibalism in two species ODE models provides a strong stabilizing effect. There is strong ecological evidence that cannibalism exists among prey as well, yet this phenomenon has been much less investigated. In the current manuscript, we investigate both the ODE and spatially explicit forms of a Holling-Tanner model, with ratio dependent functional response. We show that cannibalism in the predator provides a stabilizing influence as expected. However, when cannibalism in the prey is considered, we show that it cannot stabilise the unstable interior equilibrium in the ODE case, but can destabilise the stable interior equilibrium. In the spatially explicit case, we show that in certain parameter regime, prey cannibalism can lead to pattern forming Turing dynamics, which is an impossibility without it. Lastly we consider a stochastic prey cannibalism rate, and find that it can alter both spatial patterns, as well as limit cycle dynamics.
The Trojan Y Chromosome strategy (TYC) is a promising eradication method for biological control of nonnative species. The strategy works by manipulating the sex ratio of a population through the introduction of supermales that guarantee male offspring. In the current study, we compare the TYC method with a pure harvesting strategy. We also analyze a hybrid harvesting model that mirrors the TYC strategy. The dynamic analysis leads to results on stability of solutions and bifurcations of the model. Several conclusions about the different strategies are established via optimal control methods. In particular, the results affirm that either a pure harvesting or hybrid strategy may work better than the TYC method at controlling a nonnative species population.
Recommendations for resource managers• Where harvesting is feasible, it is as effective if not more effective than the classical TYC method. Therein managers may attempt harvesting female fish while stocking males or harvesting both male and female fishes.• Managers may attempt linear harvesting, saturating density-dependent harvesting, and unbounded density-dependent harvesting. Linear harvesting is seen to be the most effective.• We caution against the outright use of harvesting due to various density-dependent effects that may arise. To this end hybrid models that involve a combination of harvesting and TYC-type methods might be a better strategy.• One may also use harvesting as a tool in mesocosm settings to predict the efficacy of the TYC strategy in the wild.
The Trojan Y-Chromosome (TYC) strategy, an autocidal genetic biocontrol method, has been proposed to eliminate invasive alien species. In this work, we analyze the dynamical system model of the TYC strategy, with the aim of studying the viability of the TYC eradication and control strategy of an invasive species. In particular, because the constant introduction of sex-reversed trojan females for all time is not possible in practice, there arises the question: What happens if this injection is stopped after some time? Can the invasive species recover? To answer that question, we perform a rigorous bifurcation analysis and study the basin of attraction of the recovery state and the extinction state in both the full model and a certain reduced model. In particular, we find a theoretical condition for the eradication strategy to work. Additionally, the consideration of an Allee effect and the possibility of a Turing instability are also studied in this work. Our results show that: (1) with the inclusion of an Allee effect, the number of the invasive females is not required to be very low when the introduction of the sex-reversed trojan females is stopped, and the remaining Trojan Y-Chromosome population is sufficient to induce extinction of the invasive females; (2) incorporating diffusive spatial spread does not produce a Turing instability, which would have suggested that the TYC eradication strategy might be only partially effective, leaving a patchy distribution of the invasive species.
Language competition models help understand language shift dynamics, and have effectively captured how English has outcompeted various local languages, such as Scottish Gaelic in Scotland, and Mandarin in Singapore. India, with a 125 million English speakers boasts the second largest number of English speakers in the world, after the United States. The 1961-2001 Indian censuses report a sharp increase in Hindi/English Bilinguals, suggesting that English is on the rise in India. To the contrary, we claim supported by field evidence, that these statistics are inaccurate, ignoring an emerging class who do not have full bilingual competence and switch between Hindi and English, communicating via a code popularly known as "Hinglish". Since current language competition models occlude hybrid practices and detailed local ecological factors, they are inappropriate to capture the current language dynamics in India. Expanding predator-prey and sociolinguistic theories, we draw on local Indian ecological factors to develop a novel three-species model of interaction between Monolingual Hindi speakers, Hindi/English Bilinguals and Hinglish speakers, and explore the long time dynamics it predicts. The model also exhibits Turing instability, which is the first pattern formation result in language dynamics. These results challenge traditional assumptions of English encroachment in India. More broadly, the three-species model introduced here is a first step towards modeling the dynamics of hybrid language scenarios in other settings across the world.
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