Bistability induced by generalist natural enemies can reverse pest invasions

Reaction-diffusion analytical modeling of predator-prey systems has shown that specialist natural enemies can slow, stop and even reverse pest invasions, assuming that the prey population displays a strong Allee effect in its growth. Few additional analytical results have been obtained for other spatially distributed predator-prey systems, as traveling waves of non-monotonous systems are notoriously difficult to obtain. Traveling waves have indeed recently been shown to exist in predator-prey systems, but the direction of the wave, an essential item of information in the context of the control of biological invasions, is generally unknown. Preliminary numerical explorations have hinted that control by generalist predators might be possible for prey populations displaying logistic growth. We aimed to formalize the conditions in which spatial biological control can be achieved by generalists, through an analytical approach based on reaction-diffusion equations. The population of the focal prey - the invader - is assumed to grow according to a logistic function. The predator has a type II functional response and is present everywhere in the domain, at its carrying capacity, on alternative hosts. Control, defined as the invader becoming extinct in the domain, may result from spatially independent demographic dynamics or from a spatial extinction wave. Using comparison principles, we obtain sufficient conditions for control and for invasion, based on scalar bistable partial differential equations (PDEs). The searching efficiency and functional response plateau of the predator are identified as the main parameters defining the parameter space for prey extinction and invasion. Numerical explorations are carried out in the region of those control parameters space between the super-and subso-lutions, in which no conclusion about controllability can be drawn on the basis of analytical solutions. The ability of generalist predators to control prey populations with logistic growth lies in the bis-table dynamics of the coupled system, rather than in the bistability of prey-only dynamics as observed for specialist predators attacking prey populations displaying Allee effects. The consideration of space in predator-prey systems involving generalist predators with a parabolic functional response is crucial. Analysis of the ordinary differential equations (ODEs) system identifies parameter regions with monostable (extinction) and bistable (extinction or invasion) dynamics. By contrast, analysis of the associated PDE system distinguishes different and additional regions of invasion and extinction. Depending on the relative positions of these different zones, four patterns of spatial dynamics can be identified : traveling waves of extinction and invasion, pulse waves of extinction and heterogeneous stationary positive solutions of the Turing type. As a consequence, prey control is predicted to be possible when space is considered in additional situations other than those identified without considering space. The reverse situation is also possible. None of these considerations apply to spatial predator-prey systems with specialist natural enemies

Speeds of invasion for models with Allee effects

Models that describe the spread of invading organisms often assume no Allee effect. In contrast, abundant observational data provide evidence for Allee effects. In chapter 1, I study an invasion model based on an integrodifference equation with an Allee effect. I derive a general result for the sign of the speed of invasion. I then examine a special, linear-constant, Allee growth function and introduce a numerical scheme that allows me to estimate the speed of traveling wave solutions. In chapter 2, I study an invasion model based on a reaction-diffusion equation with an Allee effect. I use a special, piecewise-linear, Allee population growth rate. This function allows me to obtain traveling wave solutions and to compute wave speeds for a full range of Allee effects, including weak Allee effects. Some investigators claim that linearization fails to give the correct speed of invasion if there is an Allee effectI show that the minimum speed for a sufficiently weak Allee may be the same as that derived by means of linearization. In chapters 3 and 4, I extend a discrete-time analog of the Lotka-Volterra competition equations to an integrodifference-competition model and analyze this model by investigating traveling wave solutions. The speed of wave is calculated as a function of the model parameters by linearization. I also show that the linearization may fail to give the correct speed for the competition model with strongly interacting competitors because of the introduction of a weak Allee effect . A linear-constant approximation to the resulting Allee growth function is introduced to estimate the speed under this weak Allee effect. I also analyze the back of the wave for the competition model. Some sufficient conditions that guarantee no oscillation behind the wave are given

How predation can slow, stop or reverse a prey invasion

How predation can slow, stop or reverse a prey invasio

Density dependence in demography and dispersal generates fluctuating invasion speeds

Author Posting. © The Author(s), 2017. This is the author's version of the work. It is posted here under a nonexclusive, irrevocable, paid-up, worldwide license granted to WHOI. It is made available for personal use, not for redistribution. The definitive version was published in Proceedings of the National Academy of Sciences of the United States of America 114 (2017): 5053-5058, doi:10.1073/pnas.1618744114.Mitigating the spread of invasive species remains difficult—substantial variability in invasion speed is increasingly well-documented, but the sources of this variability are poorly understood. We report a mechanism for invasion speed variability. The combined action of density dependence in demography and dispersal can cause invasions to fluctuate, even in constant environments. Speed fluctuations occur through creation of a pushed invasion wave that moves forward not from small populations at the leading edge but instead, from larger, more established populations that “jump” forward past the previous invasion front. Variability in strength of the push generates fluctuating invasion speeds. Conditions giving rise to fluctuations are widely documented in nature, suggesting that an important source of invasion variability may be overlooked.LLS and AKS were supported by startup funds from the University of Minnesota 348 (UMN) to AKS, BL by NSF DMS-1515875, TEXM by NSF DEB-1501814, and MGN 349 by NSF DEB-1257545 and DEB-1145017

The role of Allee effect in modelling post resection recurrence of glioblastoma

Resection of the bulk of a tumour often cannot eliminate all cancer cells, due to their infiltration into the surrounding healthy tissue. This may lead to recurrence of the tumour at a later time. We use a reaction-diffusion equation based model of tumour growth to investigate how the invasion front is delayed by resection, and how this depends on the density and behaviour of the remaining cancer cells. We show that the delay time is highly sensitive to qualitative details of the proliferation dynamics of the cancer cell population. The typically assumed logistic type proliferation leads to unrealistic results, predicting immediate recurrence. We find that in glioblastoma cell cultures the cell proliferation rate is an increasing function of the density at small cell densities. Our analysis suggests that cooperative behaviour of cancer cells, analogous to the Allee effect in ecology, can play a critical role in determining the time until tumour recurrence