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This paper is concerned with the mathematical modelling of complex interactions between plant species in a harsh environment such as in the arctic. The aim of the paper is to consider whether interactions between the species change in character as environments change. For example, if the effect of climate change is to make harsh climates more benign, will this imply changes in the way species interact and affect biodiversity?We consider the interaction of two species of grass. Our model is constructed based on the notion of a summer season when the plants grow, followed by a winter season when there is no growth but when the plants are subject to the effects of events such as winter storms. Our aim is to investigate changes when the summer season is lengthened, when the climate becomes more benign, when the susceptibility of plants to damage as a result of storms is increased, and when the intensity (and number) of winter storms is varied. The models we consider provide new insights into the known behaviour of plant species interactions in such situations and a basis for further modelling and prediction.

In this paper a Branch-and-Cut algorithm, based on a formulation previously introduced by us, is proposed for the Graph Coloring Problem. Since colors are indistinguishable in graph coloring, there may typically exist many different symmetrical colorings associated with a same number of colors. If solutions to an integer programming model of the problem exhibit that property, the Branch-and-Cut method tends to behave poorly even for small size graph coloring instances. Our model avoids, to certain extent, that bottleneck. Computational experience indicates that the results we obtain improve, in most cases, on those given by the well-known exact solution graph coloring algorithm Dsatur.

The maximum cut (Max-Cut) problem has extensive applications in various real-world fields, such as network design and statistical physics. In this paper, a more practical version, the Max-Cut problem with fuzzy coefficients, is discussed. Specifically, based on credibility theory, the Max-Cut problem with fuzzy coefficients is formulated as an expected value model, a chance-constrained programming model and a dependent-chance programming model respectively according to different decision criteria. When these fuzzy coefficients are represented by special fuzzy variables like triangular fuzzy numbers and trapezoidal fuzzy numbers, the crisp equivalents of the fuzzy Max-Cut problem can be obtained. Finally, a genetic algorithm combined with fuzzy simulation techniques is designed for the general fuzzy Max-Cut problem under these models and numerical experiment confirms the effectiveness of the designed genetic algorithm.