Caswell Hal, Shyu Esther
Biology Department, Woods Hole Oceanographic Institution, Woods Hole, MA 02543, USA.
Theor Popul Biol. 2012 Dec;82(4):329-39. doi: 10.1016/j.tpb.2012.03.008.
Periodic matrix models are frequently used to describe cyclic temporal variation (seasonal or interannual) and to account for the operation of multiple processes (e.g., demography and dispersal) within a single projection interval. In either case, the models take the form of periodic matrix products. The perturbation analysis of periodic models must trace the effects of parameter changes, at each phase of the cycle, on output variables that are calculated over the entire cycle. Here, we apply matrix calculus to obtain the sensitivity and elasticity of scalar-, vector-, or matrix-valued output variables. We apply the method to linear models for periodic environments (including seasonal harvest models), to vec-permutation models in which individuals are classified by multiple criteria, and to nonlinear models including both immediate and delayed density dependence. The results can be used to evaluate management strategies and to study selection gradients in periodic environments.
周期矩阵模型经常用于描述周期性的时间变化（季节性或年际性），并用以解释单个预测期内多个过程（如种群统计学和扩散）的运行情况。在这两种情况下，模型都采用周期矩阵乘积的形式。周期模型的扰动分析必须追踪在周期的每个阶段参数变化对在整个周期内计算出的输出变量的影响。在这里，我们应用矩阵微积分来获得标量、向量或矩阵值输出变量的灵敏度和弹性。我们将该方法应用于周期性环境的线性模型（包括季节性收获模型）、个体按多个标准分类的向量置换模型以及包括即时和延迟密度依赖的非线性模型。这些结果可用于评估管理策略，并研究周期性环境中的选择梯度。
