Budget Amount *help 
¥3,500,000 (Direct Cost : ¥3,500,000)
Fiscal Year 2000 : ¥700,000 (Direct Cost : ¥700,000)
Fiscal Year 1999 : ¥700,000 (Direct Cost : ¥700,000)
Fiscal Year 1998 : ¥900,000 (Direct Cost : ¥900,000)
Fiscal Year 1997 : ¥1,200,000 (Direct Cost : ¥1,200,000)

Research Abstract 
The purpose of this research is to study the effect of time delays on persistence, which is one of the most fundamental concepts in mathematical biology, to analyze chaotic phenomena and to establish the analytic methods for mathematical models in biology with time delays. In particular, we considered (1) the chemostat models with time delays effect in growth process of the species and the recycling process of the materials ; (2) the epidemiological models with time delays in the process for susceptible individuals to become infectious after they are infected ; (3) the mathematical models with time delays in medical science for the delivery of drug directly to the macrophage by using the phagocytosis of senescent red blood cells. We investigated the effects of time delays on persistence of the above models and established the analytic methods for the global stability of the mathematical models in biology with time delays. The results obtained in this research are as follows : (1) We d
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etermined an initial age distribution of red blood cells so that the duration of the therapeutic effect is the maximum (paper 2). (2) We proved some differencedifferential inequalities and applied them for the stability analysis of nonlinear retarded or neutral functional differential systems (paper 3, 4, 6). In particular, we transformed retarded systems to neutral ones and analyzed population dynamics (paper 1, 7, 8). (3) The method obtained in 2 was applied for (2)(paper 5, 9) and its extended model (paper 10, 11, 20). (4) The same method was applied for a 2dimensional neural network and the effect of time delays on its stability (paper 12). (5) We analyzed the dynamic behaviors of mathematical models in population dynamics (difference systems) and studied the conditions for chaos to occur. We proved the system evolves to chaotic system from stable one (paper 14,18). (6) We obtained some conditions for the nonexistence of periodic solutions in the dynamical systems (paper 1517). (7) We proposed the time delayed models with the effect of the environmental hormone and analyzed their persistence and stability (paper 19). (8) We considered persistence and catastrophe for lattice models (paper 13,19). 1.モデル(3)で、薬物の効果が長く持続するような赤血球の齢分布を決定した(11の論文2). 2.差分・微分不等式を証明し、非線形の遅れ型・中立型関数微分方程式系の安定性解析に応用した(論文3,4,6).特に遅れ型を中立型に変換し安定性解析を見通しよくし、個体群力学系を解析した(論文1,7,8). 3.上記2の方法をモデル(2)の解析に用い(論文5,9),さらに拡張したモデル(2)に適用した(論文10,11,20). 4.同手法を2つのニューロンをもつニューラルネットワークに適用し、その安定性に対する時間遅れの影響を考察した(論文12). 5.個体群力学系における数理モデル(差分方程式系)の動的挙動を解析し、カオス発生条件およびその進化に対する影響を考察し、系が安定系からカオス系に進化することを示した(論文14,18). 6.時間遅れを持つ微分方程式系が周期解を持たない条件を求めた(論文1517). 7.上記モデルとは別に環境ホルモンの生物種成長への影響を考察するために時間遅れモデルを提案し,そのパーシステンスと安定性を解析した(論文19). 8.また格子モデルにおけるパーシステンスとカタストロフィーについて考察した(論文13,19). Less
