Slow-fast dynamics in a planar parasite--host model with an extinction singularity

Abstract

We study a slow-fast parasite--host model featuring a singularity at the extinction state. Using techniques from Geometric Singular Perturbation Theory (GSPT), and in particular the so-called blow-up method, we desingularize that point and reconstruct the local and global dynamics. The system we consider is in non-standard GSPT form and is characterized by a rich dynamical behavior: families of slow-fast homoclinic orbits, canard-like transitions generated by trajectories that remain close to a repelling critical manifold, and topological changes produced by infinitesimal variations of the infection rate, including the creation and destruction of an endemic equilibrium. We conclude with a numerical exploration of the model, to illustrate our analytical results.

Figures (15)

• Figure 1: Bifurcation analysis of system \ref{['eq:rescaled']} with $\mathcal{R}_d>0$ (i.e., $\alpha > 1$) and focusing on the parameter $\beta$. Solid lines correspond to stable equilibria, while dashed lines to unstable ones. (a) $u$-coordinate of $\mathbf{x_1}$ and $\mathbf{x_2}$, the red curve related to $\mathbf{x_2}$ is a straight line up to a $\mathcal{O}(\varepsilon)$-factor. Note that the $u$-coordinate of $\mathbf{x_2}$ travels a $\mathcal{O}(1)$ interval (from $1-\frac{1}{\alpha}$ to $0$) in the span of an $\mathcal{O}(\varepsilon)$ interval of $\beta$ (from $d+\varepsilon$ to $d+\varepsilon\alpha^*$; recall \ref{['eq:k']}); (b) $v$-coordinate of $\mathbf{x_1}$ and $\mathbf{x_2}$, the red curve related to $\mathbf{x_2}$ is a parabola up to a $\mathcal{O}(\varepsilon^2)$-factor. See also Figure \ref{['fig:end_eq_grad']} for four numerical simulations of this phenomenon.
• Figure 2: Nullclines of system \ref{['eq:rescaled']} for $\theta \ne 0$. In both figures $\mathcal{R}_0>1$, implying that the $v$-nullclines consist of the $u$-axis and the straight line $v=(\mathcal{R}_0 - 1)u$, which distinguish between when $\dot{v}>0$ and when $\dot{v}<0$ (otherwise the $v$-nullcline would consist of only the $u$-axis and we would have $\dot{v}<0$ for all $v>0$). On the other hand, $\dot{u}>0$ if and only if $(u,v)$ belongs to the small portion of $\Delta$ contained between the axes and the $u$-nullclines. (a) $\mathcal{R}_d > 0$ and the two cases regarding the value attained by $\mathcal{R}_0$ are shown, note that the EE $\mathbf{x_2}$ exists if and only if $\mathcal{R}_0 - 1 \in \mathcal{O}(\varepsilon)$. The $u$-nullcline inside $\Delta$ consists of two branches $L_1$ and $L_2$. (b) $\mathcal{R}_d < 0$ implying that neither $\mathbf{x_1}$ nor $\mathbf{x_2}$ exist. The $u$-nullcline consists of only one branch $L_1$.
• Figure 3: Five possible behaviors of the dynamics of system \ref{['eq:rescaled']} assuming $\theta \ne 0$ (if $\theta=0$ then the only difference lies in the fact that $L_1$ would collapse on the $v$-axis). For ease of notation, we set $m=\mathcal{R}_0-1$ when $\mathcal{R}_0>1$. (a) $\mathcal{R}_0 < 1$, $\mathcal{R}_d < 0$: the origin $\mathbf{x_0}$ is globally asymptotically stable, so the population goes extinct, through a fast piece of orbit followed by a slow piece close to the critical manifold; (b) $\mathcal{R}_0 < 1$, $\mathcal{R}_d > 0$: the parasites go extinct, as the DFE $\mathbf{x_1}$ is globally asymptotically stable. The curve $v=\gamma(u)$ defined in \ref{['eq:curve_gamma']} allows us to predict whether orbits will approach $\mathbf{x_1}$ in the slow flow from the left or from the right (see also Figure \ref{['fig:test_gamma']}); (c) $\mathcal{R}_0 > 1$, $\mathcal{R}_d < 0$: the origin $\mathbf{x_0}$ is globally asymptotically stable, so the population goes extinct, possibly after a first peak in the parasite population; (d) To be precise, this scenario corresponds to $\mathcal{R}_d > 0$, $\beta>d+\varepsilon \alpha^*$\ref{['eq:k']}; as long as this requirement is satisfied, we can allow also $\mathcal{R}_0-1 \in \mathcal{O}(\varepsilon)$. The space within the heteroclinic orbit between $\mathbf{x_0}$ and $\mathbf{x_1}$ (green) is foliated by homoclinic orbits to $\mathbf{x_0}$ (see also Figure \ref{['fig:homoclinic']} and Appendix \ref{['sec:blow-up_d']}); (e) To be precise, this scenario corresponds to $\mathcal{R}_d > 0$, $d+\varepsilon < \beta<d+\varepsilon \alpha^*$, which implies that $\mathcal{R}_0-1 \in \mathcal{O}(\varepsilon)$. In this case, the EE $\mathbf{x_2}$ is globally asymptotically stable, although orbits my pass close to the origin $\mathbf{x_0}$ and spend a long time travelling close to the repelling critical manifold before reaching a neighbourhood of $\mathbf{x_2}$ (see also Figure \ref{['fig:full_dyn']} and Appendix \ref{['sec:blow-up_e']}, and Figure \ref{['fig:end_eq_grad']} to see that $\mathbf{x_2}$ only exists for a $\mathcal{O}(\varepsilon)$-large interval of $\beta$ values).
• Figure 4: Test on the accuracy of \ref{['eq:curve_gamma']} in predicting whether an orbit will enter the slow flow with $u_\infty \lessgtr 1-1/\alpha$. Values of the parameters: $\alpha=4$, $\theta=0.5$, $\beta=0.075$, $d=0.1$ so that $\mathcal{R}_d>0$ and $\mathcal{R}_0<1$, and (a) $\varepsilon=0.001$; (b) $\varepsilon=0.0005$. The curve $v=\gamma(u)$ is plotted in black. Red points represent initial conditions corresponding to $u_\infty>1-1/\alpha$, blue dots initial conditions corresponding to $u_\infty<1-1/\alpha$. Note the increase in prediction power as $\varepsilon$ decreases.
• Figure 5: Homoclinic orbits of system \ref{['eq:rescaled']}. Values of the parameters: $\alpha=4$, $\theta=0.5$, $\beta=0.5$, and $d$ varying to showcase the two scenarios in Figure \ref{['fig:ppK1']}; note the qualitatively distinct behavior of the orbits near the $v$-axis, as well as the maximum $v$ values achieved by the orbits (panels (a) and (c) vs. panels (b) and (d)). (a) Multiple homoclinic orbits from $\mathbf{x_0}$ (the origin) to itself, for $\varepsilon=0.005$, $d=0.1$; (b) multiple homoclinic orbits from $\mathbf{x_0}$ (the origin) to itself, for $\varepsilon=0.005$, $d=0.3$; (c) comparison of homoclinic orbits (blue) close to the heteroclinic loop between $\mathbf{x_0}$ and $\mathbf{x_1}$ (blue dots) and the curve $v=\gamma(u)$ (black), recall \ref{['eq:curve_gamma']}, for $d=0.1$, $\varepsilon=0.025$, $0.01$ and $0.005$ (innermost to outermost); (d) comparison of homoclinic orbits (blue) close to the heteroclinic loop between $\mathbf{x_0}$ and $\mathbf{x_1}$ (blue dots) and the curve $v=\gamma(u)$ (black), for $d=0.3$, $\varepsilon=0.025$, $0.01$ and $0.005$ (rightmost to leftmost). Note how the approximation improves as $\varepsilon$ decreases. The distance of the orbits and the curve $\gamma$ close to $u=0$ is due to the nullcline $L_1$ not being traversed by the orbits, whereas $\gamma$ "ignores" it, as it is a curve that exists in the fast flow (recall that $L_1$ collapses on the $v$-axis as $\varepsilon \to 0$).

Theorems & Definitions (27)

• Proposition 1
• proof
• Remark 1
• Theorem 1
• proof
• Lemma 1
• proof
• Proposition 2
• proof
• Proposition 3