Modeling — Question 7

Question 7 – Solution

Strategy. Separate transfers between stages from births and deaths, then use the dominant mode and the exceptional zero eigenvalue.

Step 1: Construct the stage balances. Maturation removes juveniles and adds adults, whereas juvenile death removes individuals without adding to the adult stage. Therefore $\boxed{J'=bA-2J,\ A'=J-A}$. All coefficients have units yr$^{-1}$. Nonnegative off-diagonal rates preserve the nonnegative quadrant; the strictly positive initial populations remain positive.

Step 2: Determine the growth threshold. The characteristic equation is $\lambda^2+3\lambda+2-b=0$, so $\lambda_\pm=(-3\pm\sqrt{1+4b})/2$. Put $d=\sqrt{1+4b}$. The adult solution has dominant coefficient $$A(t)=c_+e^{\lambda_+t}+c_-e^{\lambda_-t},\qquad c_+=\frac{J_0+(1+d)A_0/2}{d}>0.$$ Thus both populations tend to zero for $0\le b<2$, approach a nonzero critical equilibrium for $b=2$, and grow without bound for $b>2$. For $b>2$, the dominant juvenile coefficient $(\lambda_++1)c_+$ is also positive.

Step 3: Solve the critical case exactly. When $b=2$, $K=J+2A=J_0+2A_0$ is conserved. Since $A'=K-3A$, $$\boxed{A=K/3+(A_0-K/3)e^{-3t},\qquad J=K/3-2(A_0-K/3)e^{-3t}.}$$ Both tend to $K/3$ thousand. The critical equilibria form the line $J=A$; there is no single equilibrium amount independent of initial data.

Step 4: Interpret the asymptotic stage ratio. Since $J=A'+A$ and $c_+>0$, division by the dominant adult mode gives $$\boxed{\lim_{t\to\infty}\frac JA=\lambda_++1 =\frac{\sqrt{1+4b}-1}{2}.}$$ For $b=0$, this limit is zero: juveniles decay at rate $2$, adults at rate $1$. The ratio describes relative stage composition even when the entire population tends to zero; it does not imply constant abundance. For $b>2$ there is no crowding term to halt growth. A carrying capacity requires additional density-dependent assumptions absent from this model.