Question

1. Chemostat with Two Predators and a Prey: Give a simple model of a chemostat with: • one nutrient • one species of micro-organism that feeds directly on the nutrient • two more species of micro-organism that feed on the first species, but do not interact directly with each other Be sure to define any parameters that are included. It is not necessary to analyze the model. 2. Sexually Transmitted Infections: Consider a population of sexually active heterosex- ual women and men, and suppose that a sexually transmitted infection is spreading in the population. Build a simple model of the situation, using four groups: susceptible women, susceptible men, infected women and infected men. (Be careful with the incidence terms.) It is not necessary to analyze the model.

Answer 1

1. Simple model of Chemostat with 2 Predators and Prey is considered by a simple food chain in chemostat with variable yield for prey population as well as the constant yield for predator population.

The Model; the prey consumes the nutrient and the predator consumes the prey, but the predator does not consume the Nutrient. Assumimng that the functional response functions are in Michaelis - Menton type.

This model establish the sufficient conditions for the existence and local stability of equilibria by using Routh - Hurwitz criteria. It is mentioned that the washout equilibrium point is a global attractor if it is only steady state.

This confirms the intuition that all the Predator and Prey species populations cannot persist if the growth rates of the Prey and Predator are relatively small and close to one. Finally, it has been analytically and numerically shown that in certain regions of the parameter space, the model with variable yield leads to oscillatory co-existence of population in Chemostat. Hence, the limit cycle behavior by the Predator and Prey population.

2.0 1.5 1.0 0.5 0.0 0.0 0.2 0.4 0.6 0.8 1.0 p,

*Note: Growth Rate vs Population

Green line above CC - Predator

Red dotted line above CRC - Prey

2. Multiple sexually transmitted infections(STIs) have threatened human health for centuries.However the popular diseases are like HIV/AIDS,gonorrhea,genital HSV-2 infection

the dynamics of the suseptable and normal

ds(s)_ds=suseptable

ds(n)_dn=normal$_images/SI-SIS.png$$SI_InsetChart_vitaloff.png$

Answer 2

1. Simple model of Chemostat with 2 Predators and Prey is considered by a simple food chain in chemostat with variable yield for prey population as well as the constant yield for predator population.

The Model; the prey consumes the nutrient and the predator consumes the prey, but the predator does not consume the Nutrient. Assumimng that the functional response functions are in Michaelis - Menton type.

This model establish the sufficient conditions for the existence and local stability of equilibria by using Routh - Hurwitz criteria. It is mentioned that the washout equilibrium point is a global attractor if it is only steady state.

This confirms the intuition that all the Predator and Prey species populations cannot persist if the growth rates of the Prey and Predator are relatively small and close to one. Finally, it has been analytically and numerically shown that in certain regions of the parameter space, the model with variable yield leads to oscillatory co-existence of population in Chemostat. Hence, the limit cycle behavior by the Predator and Prey population.

2.0 1.5 1.0 0.5 0.0 0.0 0.2 0.4 0.6 0.8 1.0 p,

*Note: Growth Rate vs Population

Green line above CC - Predator

Red dotted line above CRC - Prey

2. Multiple sexually transmitted infections(STIs) have threatened human health for centuries.However the popular diseases are like HIV/AIDS,gonorrhea,genital HSV-2 infection

the dynamics of the suseptable and normal

ds(s)_ds=suseptable

ds(n)_dn=normal$_images/SI-SIS.png$$SI_InsetChart_vitaloff.png$