The model describing the interaction between the predator and prey species is referred to as a predator-prey model. The migration of these species from one patch to another may not be instantaneous. This may be due to barriers such as a swollen river or a busy infrastructure through the natural habitat. Recent predator-prey models have either incorporated a logistic growth for the prey population or a time delay in migration of the two species. Predator-prey models with logistic growth that integrate time delays in density-dependent migration of both species have been given little attention. A Rosenzweig-MacAurther model with density-dependent migration and time delay in the migration of both species is developed and analyzed in this study. The Analysis of the model when the prey migration rate is greater than or equal to the prey growth rate, the two species will coexist, otherwise, at least one species will become extinct. A longer delay slows down the rate at which the predator and prey population increase or decrease, thus aecting the population density of these species. The prey migration due to the predator density does not greatly affect the prey density and existence compared to the other factors that cause the prey to migrate. These factors include human activities in the natural habitats like logging and natural causes like bad climatic conditions, limited food resources and overpopulation of the prey species in a patch among others.
Plant-pathogen-herbivore model describe interaction between plants, pathogens and herbivores. Plants are invaded by pathogens and herbivores while the herbivores are harvested by natural enemies such as predators and human. On the other hand, the abundance of food does not guarantee exponential growth of species who reproduce sexually and plants governed by carrying capacity. Therefore, the Allee effect may be crucial for sustaining such species. In this paper, a model of plant-pathogen-herbivore interactions that takes Allee effect and harvesting into account was developed and analyzed. The stability analysis showed that the ratio intrinsic growth rate to the environmental carrying capacity of susceptible plants must be greater than certain threshold value to raise sufficient plant biomass to sustain other species. Numerical simulations shows that all species coexist when intrinsic growth rate of plants is greater than the harvesting rate and when conversion rate of what is eaten by herbivores to newborn ones is greater than that of their natural enemies. It also shows that in the absence of susceptible plants, herbivores migrates in search of food, while others deteriorate and dies out. Furthermore, regardless of the availability of susceptible plants, the herbivores population crashes to extinction if the herbivore population is less than the lower limit required to keep the herbivores existing in the ecosystem. In the interest of conservation of all species and the environment, policy developers will greatly benefit from understanding the solutions to address clearing land for human settlement, human activities and herbivore or their natural enemies hunting. In addition, monitor species closely, especially those that reproduce sexually by establishing and maintaining the least number required to keep the species existing.
COVID-19, a novel coronavirus, is a respiratory infection which is spread between humans through small droplets expelled when a person with COVID-19 sneezes, coughs, or speaks. An SEIQR model to investigate the spread of COVID-19 was formulated and analysed. The disease free equilibrium point for formulated model was shown to be globally asymptotically stable. The endemic states were shown to exist provided that the basic reproduction number is greater than unity. By use of Routh-Hurwitz criterion and suitable Lyapunov functions, the endemic states are shown to be locally and globally asymptotically stable respectively. This means that any perturbation of the model by the introduction of infectives the model solutions will converge to the endemic states whenever reproduction number is greater than one, thus the disease transmission levels can be kept quite low or manageable with minimal deaths at the peak times of the re-occurrence.
Human papilloma Virus (HPV) is the primary infection that causes Cervical cancer. Due to the high cost of treatment for cervical cancer, protection against HPV and Cervical cancer infection may be preferable in a scarce resource settings. In this paper, a deterministic model that incorporates protection against the infection was developed and analysed. The endemic state is shown to exist provided that the reproduction number is greater than unity. Furthermore, by the use of Routh-Hurwitz criterion and suitable Lyapunov functions, Endemic Equilibrium (EE) is shown to exist provided that the reproduction number is greater than unity. By use of a suitable Lyapunov function, the endemic state was shown to be globally asymptotically stable. The effectiveness of protection is achieved if well done hence, an increase in protection leads to low disease prevalence in a population.
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