Bifurcation to locked fronts in two component reaction–diffusion systems

We study invasion fronts and spreading speeds in two component reaction–diffusion systems. Using a variation of Lin's method, we construct traveling front solutions and show the existence of a bifurcation to locked fronts where both components invade at the same speed. Expansions of the wave speed as a function of the diffusion constant of one species are obtained. The bifurcation can be sub or super-critical depending on whether the locked fronts exist for parameter values above or below the bifurcation value. Interestingly, in the sub-critical case numerical simulations reveal that the spreading speed of the PDE system does not depend continuously on the coefficient of diffusion.     

The spreading speed for a predator–prey model with one predator and two preys

We study a predator–prey model with two preys and one predator. Our main concern is the invasion process of the predator into the habitat of two aborigine preys. We consider the case when the two preys are weak competitors in the absence of predator. Under certain conditions, we are able to characterize the asymptotic spreading speed by the parameters of the model.     

Fractal dimension of the choriocarcinoma cell invasion front

The Minkowski dimension was calculated for the determination of the invasiveness of cancer cells, which were brought in close contact (confronted) to host tissues and cultivated in vitro. The confrontation cultures were histologically sectioned and stained immunohistochemically. Digital images of these sections were segmented, processed and the specimens’ boundaries were investigated. The invasion front was treated as a fractal. The value of the Minkowski dimension increased with increased invasiveness of the cells. The Minkowski dimension could be used as a quantitative measure for cancer diagnosis.     

MODELING INVASION OF PESTS RESISTANT TO Bt TOXINS PRODUCED BY GENETICALLY MODIFIED PLANTS: RECESSIVE VS. DOMINANT INVADERS

ABSTRACT. There is a growing public concern about the ecological and evolutionary consequence of the use of genetically modified organisms. We study the impact of Bt resistant pests on genetically modified Bt crops and compare exposure of Bt plants to recessive and dominant Bt resistant invaders. To simulate pest invasion we develop a conceptual reaction‐diffusion model of the Bt crop Bt susceptible insects Bt resistant insects for both the recessive and dominant pests. We show by means of computer simulations that there is a key parameter which we define as the growth number that characterizes the insects' fitness. We also show that the Bt resistant insects' invasion can lead to inhomogeneity in plant and insect spatial distributions. The plant and insect spatial patterns resulting from the Bt resistant insects' invasion are found to be dependent on the duration of the Bt resistant insect reproduction period. We compare averaged plant biomass resulting from the invasion of the dominant insects with the averaged plant biomass resulting from the invasion of the recessive insects. As a result, we show that in contrast to the recessive insects, the dominant ones initiate destruction of the plant population if the inflow of Bt susceptible insects is more than a critical value. In this case the plant biomass decays to zero. Otherwise, the plant biomass under the invasion of both the dominant and recessive insects depends on the duration of the insect reproduction period. We conclude that under invasion of dominant Bt resistant pests, the refuge strategy which has received wide acceptance in agricultural practice may not be scientifically sound practice.