Past exam of the topics in applied mathematics course of the University of Oxford / 2011 / A / 1

Write $f(N)=\alpha N^2/(\beta N^2+S)$. Its nonnegative fixed points are $N=0$ and, when ${\alpha }^2\ge 4\beta S$,Indeed a positive fixed point satisfies $\beta N^2-\alpha N+S=0$. The linear stability multiplier isAt a positive fixed point, $\beta N^2+S=\alpha N$, henceFor strict discriminant positivity, $N_{+}$ is attracting and $N_{-}$ repelling; the extinction state is superstable. At equality the positive point has multiplier one. Sinceit attracts from above and repels from below at equality. This is a semistable fixed point, whose one-sided behavior requires the nonlinear calculation.

The two positive fixed points merge whenHere $f^{\prime }(N_c)=1$, $f^{\prime \prime }(N_c)\ne 0$, and increasing $S$ decreases $f$, so this is a saddle-node bifurcation of the map. For $S<S_c$, the cobweb descends to zero below $N_{-}$, rises towards $N_{+}$ between the positive fixed points, and descends towards $N_{+}$ above it. Monotonicity of $f$ prevents a trajectory from overshooting across the attracting fixed point. Thus $N_{-}$ is an extinction threshold.

For $S>S_c$, $f(N)<N$ for every $N>0$. The iterates decrease to a nonnegative fixed point, necessarily zero. At $S=S_c$, initial states below $N_c$ tend to zero and those above tend to $N_c$. The sterile-release population threshold therefore combines bistability below the critical release with extinction above it.

The term $ru$ is unrestricted growth of species $u$, and $-r{u}^2$ represents its density-dependent self-limitation, with scaled carrying capacity one. The interaction $-ruv$ suppresses $u$ in the presence of species $v$.

Species $v$ has intrinsic growth $v$ and suppression $-\beta uv$ by $u$. Thus both interspecific effects are harmful: this is a competition model, with no self-limitation term for $v$. Its unbounded growth when $u$ disappears is a consequence of that modeling choice. The parameter $r$ is the growth-time ratio, while $\beta$ measures the effect of $u$ on $v$.

The equilibrium points areThe third is biologically interior only for $\beta >1$, and coincides with $(1,0)$ at $\beta =1$. The Jacobian matrix isAt $(0,0)$ the eigenvalues are $r,1$, so it is a repelling node. At $(1,0)$ they are $-r,1-\beta$, giving a stable node for $\beta >1$ and a saddle for $\beta <1$.

At the interior equilibrium,so coexistence is a saddle. The formal equilibrium with negative $v$ for $\beta <1$ is not a population state; there its trace is negative and determinant positive.

At $\beta =1$ the boundary equilibrium is nonhyperbolic. To determine its behavior into the positive quadrant, put $x=u-1$. A local center manifold has $x=-v+v^2/r+O(v^3)$, so $\dot{v}=v^2+O(v^3)>0$ for small positive $v$. Thus it is unstable to introduction of species $v$, though attracting along the $v=0$ axis. The boundary/interior branches undergo a transcritical bifurcation at this parameter.

The positive quadrant is invariant. The non-axis nullclines are $u+v=1$ and $u=1/\beta$. Below $u+v=1$, $u$ increases; above it, $u$ decreases. To the left of $u=1/\beta$, $v$ increases; to the right, it decreases.

For $\beta >1$, the stable manifold of the interior saddle separates trajectories tending to $(1,0)$ from those for which $u\to 0$ and $v\to \infty$. Exactly on that manifold, the trajectory tends to the saddle. For the illustrated $r=1,\beta =2$, the separating manifold is exactly $v=u$, since the vector field preserves that diagonal.

For $\beta <1$, every strictly positive initial state ultimately has $u\to 0$ and $v\to \infty$. Indeed $\dot{u}\le ru(1-u)$ implies $\mathrm{limsup}u\le 1$, so eventually $1-\beta u$ is bounded below by a positive constant. Then $v$ grows exponentially, forcing $u$ down. At $\beta =1$, a positive $v$ eventually drives $u$ below one; thereafter $v$ increases. A finite positive limit for $v$ would force $u\to 1$, inconsistent with $\dot{u}\to -rv<0$, so the same exclusion outcome holds.

There are no interior periodic orbits: the Bendixson-Dulac criterion with multiplier $1/(uv)$ givesThe phase portraits in the preceding figure show the threshold geometry and the absence of a bounded attracting coexistence state. Initial states on the axes retain the corresponding one-species dynamics.