Udriste C* and Tevy I
Received: December 21, 2018; Published: January 16, 2019
*Corresponding author: Udriste C, Department of Mathematics-Informatics, Faculty of Applied Sciences, Splaiul Independentei 313, Bucharest, 060042, Centenary of Romanian Great Union 1818-2018, Romania
A central problem in biological dynamical systems is to determine the boundaries of evolution. Although this is a general problem, we prefer to give solutions for growth of the phytoplankton. AMS Mathematical Classiftcation: 92D40, 35B36, 35Q92, 37N25.
Keywords: Phytoplankton Dynamical System; Phytoplankton Sub-strate; Phytoplankton Biomass; Intracellular Nutrient Per Biomasss
In the paper of Bernard - Gouze  one analyse a model of phytoplankton growth based on the dynamical system
where x1 means the substrate, x2 is the phytoplankton biomass and x3 is the intracellular nutrient per biomass, with the physical domain The previous dynΣamical system is non cooperative and has the equilibrium point We introduce the phytoplankton vector field of components and the maximal field line x = x(t, x0), t ∈ I, which satisfies the initial condition x(t0, x0) = x0. In order to find bounds for substrate, biomass, and intracellular nutrient per biomass, we use the techniques of optimization developed in our papers [2-6].
We use the following problem: ftnd max f (x1, x2, x3) = x1 with the restriction x=x(t,x0). We set the critical point condition . In this case It follows the relation The convenient solution (critical point) must be
The sufficient condition Hess reduces to
Since at the critical point we have the condition goes to and the convenient condition is . Theorem 2.1. Suppose that on a evolution line (field line) it exists a point x¯ at which we have
Then the phytoplankton substrate has an upper bound at this point.
Let us use the problem: ftnd max subject to x=x(t,x0).
Growth of phytoplankton: The critical point condition is it follows the relation The convenient solution (critical point) The sufficient condition HessSince at the critical point we have , the sufficient condition leads to
Theorem: Suppose that on a evolution line (field line) it exists a point at which we have
then the x2(t) component of the corresponding field line has an upper bound at this point.
In the direct alternative, we build the composite function g(x(t, x0)). The condition
reduces to the convenient solution is The condition becomes Replacing we find x2 > 0, x1 − 2 < 0. The same result is obtained as in the previous method.
Now the helping problem is: compute max with the restriction
The critical point condition is Since Δh = ∇h = (0,0,1), it follows the relation The critical point The sufficient condition Hess reduces to
Since at the critical point we have the condition goes to
Theorem: Suppose that on a evolution line (field line) it exists a point x at which we have Then the intracellular nutrient per biomass has an upper bound at this point.