Terry Alan J
Division of Mathematics, University of Dundee, Dundee, DD1 4HN, UK.
Math Biosci. 2014 Oct;256:102-15. doi: 10.1016/j.mbs.2014.08.009. Epub 2014 Sep 4.
We study a model for biological pest control (or "biocontrol") in which a pest population is controlled by a program of periodic releases of a fixed yield of predators that prey on the pest. Releases are represented as impulsive increases in the predator population. Between releases, predator-pest dynamics evolve according to a predator-prey model with some fairly general properties: the pest population grows logistically in the absence of predation; the predator functional response is either of Beddington-DeAngelis type or Holling type II; the predator per capita birth rate is bounded above by a constant multiple of the predator functional response; and the predator per capita death rate is allowed to be decreasing in the predator functional response and increasing in the predator population, though the special case in which it is constant is permitted too. We prove that, when the predator functional response is of Beddington-DeAngelis type and the predators are not sufficiently voracious, then the biocontrol program will fail to reduce the pest population below a particular economic threshold, regardless of the frequency or yield of the releases. We prove also that our model possesses a pest-eradication solution, which is both locally and globally stable provided that predators are sufficiently voracious and that releases occur sufficiently often. We establish, curiously, that the pest-eradication solution can be locally stable whilst not being globally stable, the upshot of which is that, if we delay a biocontrol response to a new pest invasion, then this can change the outcome of the response from pest eradication to pest persistence. Finally, we state a number of specific examples for our model, and, for one of these examples, we corroborate parts of our analysis by numerical simulations.
我们研究了一种生物害虫控制（或“生物防治”）模型，在该模型中，害虫种群通过定期释放固定产量的捕食害虫的天敌来控制。释放被表示为天敌种群的脉冲式增加。在两次释放之间，天敌 - 害虫动态根据具有一些相当一般性质的捕食者 - 猎物模型演变：在没有捕食的情况下，害虫种群呈逻辑斯谛增长；天敌功能反应要么是贝丁顿 - 迪安吉利斯型，要么是霍林II型；天敌人均出生率以天敌功能反应的常数倍为上限；并且天敌人均死亡率允许随着天敌功能反应而降低，随着天敌种群增加而增加，不过也允许其为常数的特殊情况。我们证明，当天敌功能反应是贝丁顿 - 迪安吉利斯型且天敌捕食能力不足时，那么生物防治计划将无法将害虫种群减少到特定的经济阈值以下，无论释放的频率或产量如何。我们还证明，我们的模型具有害虫根除解，只要天敌捕食能力足够强且释放足够频繁，该解在局部和全局都是稳定的。奇怪的是，我们确定害虫根除解可以是局部稳定但不是全局稳定的，其结果是，如果我们延迟对新害虫入侵的生物防治反应，那么这可能会将反应结果从害虫根除变为害虫持续存在。最后，我们给出了我们模型的一些具体例子，并且对于其中一个例子，我们通过数值模拟证实了我们分析的部分内容。
