Multitype contact process with sterile states

TL;DR

We study a spatial variant of the multitype contact process on $\mathbb{Z}^d$ where fertile individuals of type $i$ reproduce at rate $\lambda_i$, and offspring are fertile with probability $p_i$ and sterile with probability $1-p_i$. The work contrasts nonspatial mean-field predictions, which state survival whenever $\lambda_i p_i>1$ and single-type competition is governed by the largest $\lambda_i p_i$, with spatial results showing extinction for small $p$ even when $\lambda p$ is large, and nuanced winner/coexistence behavior that depends on spatial structure. The authors prove extinction when $\lambda p \leq \lambda_c$ via a graphical coupling to the basic contact process, establish survival for any $\lambda>\lambda_c$ when $p$ is near 1, and show extinction for $p<1/(4d)$ through a subcritical Galton–Watson bound; in the two-type setting, they identify conditions under which type 1 wins even when type 2 has arbitrarily large $\lambda_2$. These results highlight how spatial local interactions qualitatively alter competition outcomes and demonstrate that the simple product $\lambda p$ ceases to fully predict competitiveness in the spatial model.

Abstract

This paper considers a natural variant of the $d$-dimensional multitype contact process in which individuals can be fertile or sterile. Fertile individuals of type $i$ give birth to an offspring of their own type at rate $λ_i$, the offspring being fertile with probability $p_i$ and sterile with probability $1 - p_i$, whereas sterile individuals can't give birth. Offspring are sent to one of the neighbors of their parent's location and take place in the system if and only if the target site is empty. All the individuals die at rate one regardless of their type and regardless of whether they are fertile or sterile. Our main results show some qualitative disagreements between the spatial model and its nonspatial mean-field approximation that are more pronounced when the probability $p_i$ is small. More precisely, for the mean-field model, in the presence of only one type, survival occurs when $λ_i p_i > 1$, and in the presence of two types, the type with the largest $λ_i p_i$ wins. In contrast, though the analysis of the spatial model shows a similar behavior when $p_i$ is close to one, in the presence of only one type, extinction always occurs when $p_i < 1/4d$. Similarly, a type with $λ_i > λ_c =$ critical value of the contact process and $p_i = 1$ is more competitive than a type with $λ_i$ arbitrarily large but $p_i < 1/4d$, showing that the product $λ_i p_i$ no longer measures the competitiveness. These results underline the effects of space in the form of local interactions.

Figures (8)

• Figure 1: Snapshots at time 1$00$ of the contact process with a sterile state. The black sites represent the fertile individuals, the gray sites the sterile individuals, and the white sites the empty sites.
• Figure 2: Phase structure of the single-type model. The dashed curve with equation $\lambda p = 1$ separates the extinction phase from the survival phase for the mean-field model. The solid black curve shows the phase transition of the spatial model. This curve lies above the gray curve $\lambda p = \lambda_c$ according to Theorem \ref{['th:coupling']} and above the gray line $p = 1/4d$ according to Theorem \ref{['th:extinction']}. This curve also converges to one as $\lambda \downarrow \lambda_c$ according to Theorems \ref{['th:survival']}--\ref{['th:extinction']}.
• Figure 3: Solution curves of the mean-field model \ref{['eq:mf-01']}. The horizontal axis represents the density of fertile individuals, while the vertical axis represents the density of sterile individuals. In both pictures $\lambda p = 2 > 1$ so all the solution curves starting from $u_+ \neq 0$ converge to the unique interior fixed point $Q$. The limiting density of individuals is the same in both pictures, but the limiting density of fertile individuals is smaller and the limiting density of sterile individuals larger in the second picture because $p$ is smaller.
• Figure 4: Projection of the solution curves of the mean-field model \ref{['eq:mf-07']}. The horizontal axis represents the density of type 1 individuals, while the vertical axis represents the density of type 2 individuals. In both pictures, the solution curves start (and remain) in the invariant subspace $\Gamma_2$. In the left picture, $\lambda_2 p_2 > \lambda_1 p_1 > 1$ so $Q_2$ is locally stable, while $Q_1$ is unstable: type 2 wins. In the right picture, $\lambda_1 p_1 = \lambda_2 p_2 = 2 > 1$ so the set of interior fixed points consists of line segment connecting $Q_1$ and $Q_2$: both types coexist.
• Figure 5: Graphical representation of the single-type process. The light gray lines represent the empty sites (0), the dark gray lines the sterile individuals ($- 1$), and the black lines the fertile individuals ($+ 1$).

Theorems & Definitions (17)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Lemma 5
• Proposition 6
• Proposition 7
• Lemma 8
• Lemma 9
• Lemma 10