C$.$S$.$ Holling $(1978)$ proposed the following model of the forest budworm system:

$\frac{dN}{dt}=RN(1-\frac{N}{K})-BP(\frac{N^2}{A^2+N^2}).$

The first term on the right$-$hand side is just logistic growth for the spruce budworm population, $N$ is the density of the budworms. $R$ is the maximum growth rate of the

spruce budworm population, $K$ is the carrying capacity of the forest $($here we can assume that it is directly proportional to the leaf biomass$).$

The second term is a functional response of the predator birds times the number of predators $P,B$ is the maximum predation rate of an individual bird $($on average$),$ and $A$ is the budworm population when the predation rate is at half the maximum.

Part $1$

The forest biomass is directly proportional to $K,$ which in this model is con$-$stant. Give a reasonable rationale for this, i$.$e$.,$ what assumptions do you think Ludwig et al$.$ used to justify this.

The predator $($bird$)$ population is also taken as a constant. Explain.

Part $2$

In order to more easily analyze the model we first nondimensionalize it$.$ We can write without loss of generality the following dynamically equivalent equation:

$\frac{dx}{dt}=rx(1-\frac{x}{k})-(\frac{x^2}{1+x^2}).$

Notice that we now have to deal with only two parameters. Also, all the pa$-$rameters and variables are dimensionless, which means that we have to knowonly relative values of things rather than absolute values, which may depend on the scale of measurement and other factors. What are the appropriate transformations of the variables $N$ and $t$ such that the model has only the two parameters $r$ and $k,$ and what are $r$ and $k$ in terms of the original parameters, and $P?$

Interpret in general terms $($in the sense of "something relative to somethingelse"$)$ the rescaled variables $x$ and $t$ and parameters $r$ and $k.[$Hint: What has"disappeared" from the functional response?$]$