Foxes eat rabbits and are not interested in eating carrots, but rabbits eat carrots

(Food web] (a) Foxes eat rabbits and are not interested in eating carrots, but rabbits eat carrots. You want to understand how the abundance of carrots impact the number of foxes, assuming that the resource is limited (i.e., rabbits consumption decreases the volume of carrots). In the absence of carrots, rabbits die, and in the absence of rabbits, foxes die, both at a specific rate. Make a flow diagram and write the generalized prey-predator equations associated with this competition. (b) Consider now a population of another type of rabbits, that has no predator, and eat carrots or grass, two limited resources with distinct reproduction rates, and impacting with a distinct coefficient the rabbit reproduction (e.g., rabbits need to eat twice more grass than carrots to reproduce). In the absence of both resources, rabbits cannot survive. Make a diagram and write the equations for this food web.

The Correct Answer and Explanation is :

Part (a): Foxes, Rabbits, and Carrots – Prey-Predator Model

In this food web, the relationships can be broken down as follows:

1. Carrots serve as the primary resource for rabbits, which consume them.
2. Rabbits are prey for foxes, so the number of foxes depends on the number of rabbits.
3. Foxes do not consume carrots directly, so the abundance of carrots indirectly affects fox populations by influencing the rabbit population.

Let’s denote:

• ( C(t) ): Carrot population at time ( t )
• ( R(t) ): Rabbit population at time ( t )
• ( F(t) ): Fox population at time ( t )

Flow Diagram Description

• Carrots increase or decrease based on how much the rabbits consume them. The carrot population decreases with the rate ( \alpha R(t) ), where ( \alpha ) is the rate at which rabbits consume carrots.
• Rabbits are born at a rate proportional to the available carrots, and they die at a rate proportional to their population.
• Foxes feed on rabbits, which will increase the fox population at a rate proportional to the rabbit population. Foxes also die at a natural rate.

Prey-Predator Equations:

1. Carrots:
   [
   \frac{dC}{dt} = – \alpha R(t) C(t)
   ]
   The carrots decrease as rabbits consume them.
2. Rabbits:
   [
   \frac{dR}{dt} = \beta R(t) C(t) – \gamma R(t) – \delta R(t) F(t)
   ]
   Here, ( \beta ) is the rate at which rabbits reproduce based on carrot availability, ( \gamma ) represents natural death, and ( \delta ) is the rate at which foxes predate rabbits.
3. Foxes:
   [
   \frac{dF}{dt} = \mu R(t) F(t) – \nu F(t)
   ]
   Foxes grow in population by eating rabbits, at a rate proportional to both rabbit and fox populations (( \mu )), and naturally die at a rate ( \nu ).

Part (b): Rabbits with Two Resources (Carrots and Grass)

In this case, rabbits have two resources: carrots and grass. The rabbits have different reproductive rates based on the type of resource they consume.

Let’s denote:

• ( C(t) ): Carrot population at time ( t )
• ( G(t) ): Grass population at time ( t )
• ( R(t) ): Rabbit population at time ( t )

Flow Diagram Description

• Rabbits eat both carrots and grass. The availability of these resources directly impacts their reproduction.
• Carrots and grass both contribute to rabbit reproduction, but grass is more important, requiring twice as much grass as carrots to reproduce.
• Rabbits die if neither resource is available in sufficient amounts.

Equations for This Web:

1. Rabbits:
   [
   \frac{dR}{dt} = \alpha \left( \frac{C(t)}{2} + G(t) \right) R(t) – \beta R(t)
   ]
   Here, ( \alpha ) represents the efficiency of each resource in supporting reproduction, and ( \beta ) is the natural death rate of rabbits.
2. Carrots:
   [
   \frac{dC}{dt} = – \delta R(t) C(t)
   ]
   Similar to the previous model, carrots are consumed by rabbits at a rate proportional to the rabbit population.
3. Grass:
   [
   \frac{dG}{dt} = – \theta R(t) G(t)
   ]
   Grass is also consumed by rabbits, and the decrease in grass is proportional to rabbit population.

I will now generate the flow diagrams and equations for you.

Here is the flow diagram illustrating the food web with carrots, rabbits, and foxes. The diagram highlights the relationships between the populations: how carrots are consumed by rabbits, and how foxes prey on rabbits. It also includes feedback loops that show how the populations influence each other.

Let me know if you need any further explanations or modifications!