2. A garden is home to a complicated ecosystem. We will consider a garden with three species: aphids, ladybugs, and caterpillars/butterflies. You have been observing the garden, and have identified the following important interactions in this ecosystem: Butterflies have abundant food resources, that do not need to be modelled. When butterflies reproduce, they produce caterpillars. Caterpillars mature at a constant rate, becoming butterflies after a fixed duration. Both caterpillars and butterflies can die due to external factors. Caterpillars and aphids compete for food. The reproduction rate of aphids and the death rate of caterpillars is influenced by this competition. Ladybugs predate upon aphids. Ladybugs like their space: if they feel like that it's getting crowded, they'll move to a different garden. The ladybugs in the garden have a population cap. Both predation and the population cap affect ladybug birth rates. This question requires you to do the following: (a) Write a mathematical ODE model for the population of caterpillars, aphids, butterflies, and ladybugs. This model should contain four differential equations. Feel free to make whatever assumptions you wish, but you should explain any assumptions that you make. (b) Non-dimensionalise the model. Use the initial value of each population to scale the four population variables, and make a sensible choice to non-dimensionalise time. Challenge question (4 marks out of the 15 available in Q2): (c) A disease sweeps through the aphid population, creating two groups: uninfected and infected aphids. The disease is passed on through contact between infected aphids and uninfected aphids. After a fixed period of time the infected aphids recover, but they can be reinfected. Ladybugs have delicate tastes, and will not eat infected aphids. All aphids still compete with caterpillars for food, and the disease does not affect aphid reproduction. Incorporate the effects of the disease into your model from (a). 2. A garden is home to a complicated ecosystem. We will consider a garden with three species: aphids, ladybugs, and caterpillars/butterflies. You have been observing the garden, and have identified the following important interactions in this ecosystem: Butterflies have abundant food resources, that do not need to be modelled. When butterflies reproduce, they produce caterpillars. Caterpillars mature at a constant rate, becoming butterflies after a fixed duration. Both caterpillars and butterflies can die due to external factors. Caterpillars and aphids compete for food. The reproduction rate of aphids and the death rate of caterpillars is influenced by this competition. Ladybugs predate upon aphids. Ladybugs like their space: if they feel like that it's getting crowded, they'll move to a different garden. The ladybugs in the garden have a population cap. Both predation and the population cap affect ladybug birth rates. This question requires you to do the following: (a) Write a mathematical ODE model for the population of caterpillars, aphids, butterflies, and ladybugs. This model should contain four differential equations. Feel free to make whatever assumptions you wish, but you should explain any assumptions that you make. (b) Non-dimensionalise the model. Use the initial value of each population to scale the four population variables, and make a sensible choice to non-dimensionalise time. Challenge question (4 marks out of the 15 available in Q2): (c) A disease sweeps through the aphid population, creating two groups: uninfected and infected aphids. The disease is passed on through contact between infected aphids and uninfected aphids. After a fixed period of time the infected aphids recover, but they can be reinfected. Ladybugs have delicate tastes, and will not eat infected aphids. All aphids still compete with caterpillars for food, and the disease does not affect aphid reproduction. Incorporate the effects of the disease into your model from (a)