@article{ackleh_hu_2007, title={Comparison between stochastic and deterministic selection-mutation models}, volume={4}, ISSN={["1551-0018"]}, DOI={10.3934/mbe.2007.4.133}, abstractNote={We present a deterministic selection-mutation model with a discrete trait variable. We show that for an irreducible selection-mutation matrix in the birth term the deterministic model has a unique interior equilibrium which is globally stable. Thus all subpopulations coexist. In the pure selection case, the outcome is known to be that of competitive exclusion, where the subpopulation with the largest growth-to-mortality ratio will survive and the remaining subpopulations will go extinct. We show that if the selection mutation matrix is reducible, then competitive exclusion or coexistence are possible outcomes. We then develop a stochastic population model based on the deterministic one. We show numerically that the mean behavior of the stochastic model in general agrees with the deterministic one. However, un like the deterministic one, if the differences in the growth-to-mortality ratios are small in the pure selection case, it cannot be determined a priori which subpopulation will have the highest probability of surviving and winning the competition.}, number={2}, journal={MATHEMATICAL BIOSCIENCES AND ENGINEERING}, author={Ackleh, Azmy S. and Hu, Shuhua}, year={2007}, month={Apr}, pages={133–157} }
 @article{ackleh_deng_hu_2007, title={On a nonlinear size-structured phytoplankton-zooplankton aggregation model}, volume={14}, number={2}, journal={Dynamics of Continuous, Discrete & Impulsive Systems. Series A, Mathematical Analysis}, author={Ackleh, A. S. and Deng, K. and Hu, S. H.}, year={2007}, pages={265–285} }
 @article{ackleh_deng_ito_thibodeaux_2006, title={A structured erythropoiesis model with nonlinear cell maturation velocity and hormone decay rate}, volume={204}, ISSN={["1879-3134"]}, DOI={10.1016/j.mbs.2006.08.004}, abstractNote={We develop a quasilinear structured model that describes the regulation of erythropoiesis, the process in which red blood cells are developed. In our model, the maturation velocity of precursor cells is assumed to be a function of the erythropoietin hormone, and the decay rate of this hormone is assumed to be a function of the number of precursor cells, unlike other models which assume these parameters to be constants. Existence-uniqueness results are established and convergence of a finite difference approximation to the unique solution of the model is obtained. The finite difference scheme is then used to investigate the effects of these nonlinear parameters on the model dynamics. Our results show that a velocity of precursor cells maturation rate which is an increasing function of the hormone level and a decay rate of the hormone which is an increasing function of the number of precursor cells have a stabilizing effect on the dynamics of the model. While assuming that one parameter is a function and letting the other be a constant stabilizes the oscillations in the mature cells level, the effect is more significant when both parameters are taken to be functions. A study of robustness with respect to the forms of these functions and parameter sensitivity is also carried out.}, number={1}, journal={MATHEMATICAL BIOSCIENCES}, author={Ackleh, Azmy S. and Deng, Keng and Ito, Kazufumi and Thibodeaux, Jeremy}, year={2006}, month={Nov}, pages={21–48} }
 @article{ackleh_banks_deng_2002, title={A finite difference approximation for a coupled system of nonlinear size-structured populations}, volume={50}, ISSN={["1873-5215"]}, DOI={10.1016/S0362-546X(01)00780-5}, abstractNote={Abstract : We study a quasilinear nonlocal hyperbolic initial-boundary value problem that models the evolution of N size-structured subpopulations competing for common resources. We develop an implicit finite difference scheme to approximate the solution of this model. The convergence of this approximation to a unique bounded variation weak solution is obtained. The numerical results for a special case of this model suggest that when subpopulations are closed under reproduction, one subpopulation survives and the others go to extinction. Moreover, in the case of open reproduction, survival of more than one population is possible.}, number={6}, journal={NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS}, author={Ackleh, AS and Banks, HT and Deng, K}, year={2002}, month={Sep}, pages={727–748} }
 @article{ackleh_fitzpatrick_1997, title={Modeling aggregation and growth processes in an algal population model: Analysis and computations}, volume={35}, ISSN={["0303-6812"]}, DOI={10.1007/s002850050062}, abstractNote={Aggregation, the formation of large particles through multiple collision of smaller ones is a highly visible phenomena in oceanic waters which can control material flux to the deep sea. Oceanic aggregates more than 1 cm in diameter have been observed and are frequently described to consist of phytoplankton cells as well as other organic matter such as fecel pellets and mucus nets from pteropods. Division of live phytoplankton cells within an aggregate can also increase the size of aggregate (assuming some daughter cells stay in the aggregate) and hence could be a significant factor in speeding up the formation process of larger aggregate. Due to the difficulty of modeling cell division within aggregates, few efforts have been made in this direction. In this paper, we propose a size structured approach that includes growth of aggregate size due to both cell division and aggregation. We first examine some basic mathematical issues associated with the development of a numerical simulation of the resulting algal aggregation model. The numerical algorithm is then used to examine the basic model behavior and present a comparison between aggregate distribution with and without division in aggregates. Results indicate that the inclusion of a growth term in aggregates, due to cell division, results in higher densities of larger aggregates; hence it has the impact to speed clearance of organic matter from the surface layer of the ocean.}, number={4}, journal={JOURNAL OF MATHEMATICAL BIOLOGY}, author={Ackleh, AS and Fitzpatrick, BG}, year={1997}, month={Mar}, pages={480–502} }
 @article{ackleh_1997, title={Parameter estimation in a structured algal coagulation-fragmentation model}, volume={28}, ISSN={["0362-546X"]}, DOI={10.1016/0362-546X(95)00195-2}, number={5}, journal={NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS}, author={Ackleh, AS}, year={1997}, month={Mar}, pages={837–854} }