综合运用偏微分方程、随机微分方程和对策论方法来研究生物种群适应行为动力学性态，把适应动力学研究范围从自治常微分方程发展到具有周期环境的模型、具有随机环境的模型，从数学研究方法上获得一些重要进展和创新；传统的适应动力学要求生态学演化和生物特性演化时间尺度上的分离，我们将去掉这一较为严厉的限制，以便能研究种群快速适应的情形；数学建模以适合度为基础，引入非局部进化扩散，让扩散策略不但考虑收益因素，也考虑其代价，还反映种群大范围的响应能力，数学上需要对解全局存在和种群基本再生数的计算方法创新；当种群跟随气候变化随斑块迁移时，我们将研究捕食与被捕食模型，把单调系统的研究推广到非单调情形；传统的适应动力学假定生态学参数是常数，我们将研究周期生态环境下生物种群适应行为演化的规律。

Adaptive dynamics of biological populations are studied by partial differential equations, stochastic differential equations and game theory. Models for adaptive behaviors are extended from autonomous ordinary differential equations to the models with periodic environment and stochastic perturbations. New mathematical approaches are achieved through the researches. Classical theory of adaptive dynamics needs the separation of ecological time scale with evolutionary time scale, which will be removed to include rapid evolution. Mathematical models are established in terms of fitness of population that incorporates both benefit and cost. Nonlocal diffusions are also considered to mimic the long range responses of individuals to environment. Mathematics for the global existence of solutions of models and computation of basic reproduction number of populations are developed.