Can a barrier zone stop invasion of a population?
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Mathematical Biology
Can a barrier zone stop invasion of a population? Bingtuan Li1
· Minghua Zhang1 · Bradley Coffman1
Received: 3 September 2019 / Revised: 12 August 2020 / Published online: 2 October 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We consider an integro-difference model to study the effect of a stationary barrier zone on invasion of a population with a strong Allee effect. It is assumed that inside the barrier zone a certain proportion of the population is killed. A Laplace dispersal kernel is used in the model. We provide a formula for the critical width L ∗ of barrier zone. We show that when a barrier zone is set at the front of a population, if the width of barrier zone is bigger than L ∗ then the barrier zone can stop the population invasion, and if the width of barrier zone is less than L ∗ then the population crosses the barrier zone and eventually occupies the entire space. The results are proven by establishing the existence and attractivity of three types of equilibrium solutions. The mathematical proofs involve phase plane analysis and comparison. Keywords Integro-difference equation · Barrier zone · Allee effect · Equilibrium · Convergence Mathematics Subject Classification 92D40 · 92D25
1 Introduction Barrier zones are used against insect species. A barrier zone can be viewed as an area at the front of a population invasion where eradication or suppression activity is performed. Tactics for managing barrier zones involve use of traps baited with attractants such as pheromones or host plant volatiles (Sharov et al. 1998), application of insecticides (Hajek and Tobin 2010), removal or (chemical or microbial) pesticide treatments of host plants (Haack et al. 2010; McCullough et al. 2015; Mercader et al. 2015; Sharov
B. Li and M. Zhang were partially supported by the National Science Foundation under Grant DMS-1515875, and B. Li and B. Coffman were partially supported by the National Science Foundation under Grant DMS-1951482.
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Bingtuan Li [email protected] Department of Mathematics, University of Louisville, Louisville, KY 40292, USA
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et al. 1998; Suckling et al. 2007), mating disruption treatments (Scruggs 1975; Suckling et al. 2014; Tobin and Blackburn 2007), predator augmentation (Hilker et al. 2005), or sterile insect releases (Knipling 1978; Suckling et al. 2007). They can be basically classified as density-independent with a certain proportion of the population killed, or density-dependent with growth inversely dependent on density and scarcity playing a role. A barrier zone can be either stationary or moving (Alexei and Andrew 1998). The barrier zone concept of slowing or stopping range expansion has been implemented against a variety of species (Sharov 2004). In 1923, a barrier zone was established along the Hudson River to prevent the spread of the gypsy moth (Lymantria dispar (L.)) (McFadden and McManus 1991). The USDA Forest Service has been conducting the Slow-the-Spread program to slow the expansion of gypsy
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