124 The UMAP Journal 30.2 (2009) Altogether, we have an interdependent and mutually-restricting mathe- matical model of the three populations: 示1()=a1(r1-入12) 示2()=m2(72+21-03), () Since this system of differential equations has no analytic solution, weneed to use Matlab to get its numerical solution. Ecologists point out that a periodic solution cannot be observed in most balanced ecosystems; in a balanced ecosystem, there is an equilibrium In addition, some ecologists think that the long-existing and periodically changing balanced ecosystems in nature tend toward a stable equilibrium; that is, if the system diverges from the former periodic cycle because of disturbance, an internal control mechanism will restore it. However, the periodically-changingstate describedby the Volterramodelisnon-structured stability, and even subtle adjustments to the parameters will change the pe riodic solution So we improve the model by letting the alga species follow logistic growth if in isolation 立1()=nz1(1 N where N, is the maximum population of the alga species allowed by the Population 2, so the model for the algae species is i1()=m11 where N2 is the maximum capacity of the species in Population 2 and o refers to the quantity ofthealgae(compared to N)eatenby theunitquantity species in Population 2(compared to N2) without the algae, the species in Population 2 will perish; let its death rate be r2, so that in isolation we will have (+) The algae provide food for Population 2, so we should add that effect; and the growth of the species in Population 2 is also influenced by internal blocking action; so we get (=(1-+mR)124 The UMAP Journal 30.2 (2009) Altogether, we have an interdependent and mutually-restricting mathe￾matical model of the three populations: ::l(t) = XI(T1 - AX) &2()= Xr2 (-r 2 + A2X1 - AX) &3 (t) = X3(-r 3 + A3X2)- Since this system of differential equations has no analytic solution, we need to use Matlab to get its numerical solution. Ecologists point out that a periodic solution cannot be observed in most balanced ecosystems; in a balanced ecosystem, there is an equilibrium. In addition, some ecologists think that the long-existing and periodically￾changing balanced ecosystems in nature tend toward a stable equilibrium; that is, if the system diverges from the former periodic cycle because of disturbance, an internal control mechanism will restore it. However, the periodically-changingstate describ edby the Volterramodelis non-structured stability, and even subtle adjustments to the parameters will change the pe￾riodic solution. So we improve the model by letting the alga species follow logistic growth if in isolation: =rix, 1 - where N1 is the maximum population of the alga species allowed by the environmental resources. The alga species provides food for the species of Population 2, so the model for the algae species is N1 X2' where N2 is the maximum capacity of the species in Population 2 and o1 refers to the quantity of the algae (compared to N1) eatenbythe unit quantity species in Population 2 (compared to N2). Without the algae, the species in Population 2 will perish; let its death rate be r 2, so that in isolation we will have: &2(t) = -r 2x 2. The algae provide food for Population 2, so we should add that effect; and the growth of the species in Population 2 is also influenced by internal blocking action; so we get &2 (t) = r2X2 (1_ -X2 + 02 X