A vector Allee effect in mosquito dynamics

Abstract

We consider a recently introduced model of mosquito dynamics that includes mating and progression through breeding, questing and egg-laying stages of mosquitoes using human and other vertebrate sources for blood meals. By exploiting a multiscale character of the model and recent results on their uniform-in-time asymptotics, we derive a simplified monotone model with the same long-term dynamics. Using the theory of monotone dynamical systems, we show that for a range of parameters, the latter displays Allee-type dynamics; that is, it has one extinction and two positive equilibria ordered with respect to the positive cone $\mathbb R_+^7$, with the extinction and the larger equilibrium being attractive and the middle one unstable. Using asymptotic analysis, we show that the original system also displays this pattern.

Figures (5)

• Figure 1: Compartments of the model.
• Figure 2: Saddle-node bifurcation in the model.
• Figure 3: 2-dimensional representation of the case ${N} > {N}^*$. The horizontal axis represents the aquatic lifeforms, while all other coordinates are combined on the vertical axis. The box shaded in orange represents the basin of attraction for the steady state $\boldsymbol{x}^*\left(A_{2} \right)$, while the box shaded in blue represents the basin of attraction of the trivial steady state.
• Figure 4: The quality of approximation for initial conditions in the basin of attraction of the stable equilibrium $\boldsymbol x(142.943)$ for different values of $\epsilon$; the case $\epsilon=0$ corresponds to the solution of \ref{['eq:fullsystemmodel1']}.
• Figure 5: The parameter values used here are the same as those in Figure \ref{['fig:stablenonzero']}. Without altering these parameter values, we modify the initial conditions so that they fall within the basin of attraction of the trivial steady state. The trajectory for $\epsilon=0$ corresponds to the solution of \ref{['eq:fullsystemmodel1']}.

Theorems & Definitions (9)

• Lemma 3.1
• Theorem 3.1
• Remark 3.1
• Theorem 3.2: Existence of Steady State Solutions
• Theorem 3.3
• Lemma 3.2
• Theorem 3.4
• Corollary 3.1
• Theorem 4.1