Ackleh Azmy S, Farkas József Z, Li Xinyu, Ma Baoling
a Department of Mathematics , University of Louisiana at Lafayette , Lafayette , LA 70504 , USA.
J Biol Dyn. 2015;9 Suppl 1:2-31. doi: 10.1080/17513758.2014.923117. Epub 2014 Jun 3.
We consider a size-structured population model where individuals may be recruited into the population at different sizes. First- and second-order finite difference schemes are developed to approximate the solution of the model. The convergence of the approximations to a unique weak solution is proved. We then show that as the distribution of the new recruits become concentrated at the smallest size, the weak solution of the distributed states-at-birth model converges to the weak solution of the classical Gurtin-McCamy-type size-structured model in the weak* topology. Numerical simulations are provided to demonstrate the achievement of the desired accuracy of the two methods for smooth solutions as well as the superior performance of the second-order method in resolving solution-discontinuities. Finally, we provide an example where supercritical Hopf-bifurcation occurs in the limiting single state-at-birth model and we apply the second-order numerical scheme to show that such bifurcation also occurs in the distributed model.
我们考虑一个大小结构的种群模型，其中个体可以在不同大小进入种群。开发了一阶和二阶有限差分格式来逼近该模型的解。证明了这些逼近收敛到唯一的弱解。然后我们表明，随着新进入者的分布集中在最小尺寸，出生时分布状态模型的弱解在弱*拓扑中收敛到经典Gurtin-McCamy型大小结构模型的弱解。提供了数值模拟，以证明两种方法对于光滑解达到所需精度，以及二阶方法在解决解的不连续性方面的优越性能。最后，我们给出一个例子，其中在极限单出生状态模型中发生超临界霍普夫分岔，并且我们应用二阶数值格式表明这种分岔也发生在分布模型中。
