A new coexistence result for competing contact processes

Abstract

Neuhauser [Probab. Theory Related Fields 91 (1992) 467--506] considered the two-type contact process and showed that on $\mathbb{Z}^2$ coexistence is not possible if the death rates are equal and the particles use the same dispersal neighborhood. Here, we show that it is possible for a species with a long-, but finite, range dispersal kernel to coexist with a superior competitor with nearest-neighbor dispersal in a model that includes deaths of blocks due to ``forest fires.''Comment: Published at http://dx.doi.org/10.1214/105051606000000132 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Coexistence for a multitype contact process with seasons

“…In some cases, the presence of two types of space is not immediately obvious. Chan and Durrett (2006) considered a generalization of the multitype contact process in which species 1 is a good competitor with a short range dispersal kernel, whereas species 2 has a long range dispersal kernel which makes it a good colonizer. In this situation, occurrences of forest fires (i.e., removal of all the individuals contained in a large square) at an appropriate rate allows species 2 to survive by migrating to newly created patches.…”

Coexistence for a population model with forest fire epidemics

“…Remark 1.3. A related model was studied by Chan and Durrett [CD06], who proved coexistence for the two-type, continuous time contact processes in Z 2 with the addition of a different type of forest fires, which act by killing all individuals (regardless of their type, and regardless of whether they are connected) within blocks of a certain size. They showed that if the weaker competitor has a larger dispersal range then it is possible for the two species to coexist in the model with forest fires; this contrasts with Neuhauser's result [Neu92] for the model without forest fires for which such coexistence is impossible.…”

Survival and coexistence for a multitype contact process

“…The model in [5] exhibits "global" coexistence with unequal rates, in that P (| 1 ξ t | ≥ 1, | 2 ξ t | ≥ 1 ∀t > 0) > 0 for every initial configuration with | 1 ξ 0 | ≥ 1, | 2 ξ 0 | ≥ 1. Coexistence results are proved in [1] for the multitype contact process on Z d with long-range interactions, or an additional "death" mechanism. Theorem 1 may give the first "local" coexistence result for a nearest-neighbor interaction model with equal death rates and unequal birth rates.…”

“…This means that when λ 1 < λ 2 and both rates lie in (λ * , λ * ], the 2's are not strong enough to drive the 1's from bounded regions of the tree, and coexistence is possible. Two-type competition models have been studied by many others; see [1,3] and [5] for instance. The model in [5] exhibits "global" coexistence with unequal rates, in that P (| 1 ξ t | ≥ 1, | 2 ξ t | ≥ 1 ∀t > 0) > 0 for every initial configuration with | 1 ξ 0 | ≥ 1, | 2 ξ 0 | ≥ 1.…”