Optimal control of biological resources with innovations

In the given work the approach to the description of active innovative processes in mathematical models of optimum resource management by economic criteria, such as renewed and not renewed natural resources, agricultural and biotechnological populations develops, etc. the active are understood as the operated innovative processes demanding certain expenses. In a reality as those it is difficult enough to allocate these processes from the general dynamics of functioning of concrete objects in modern conditions of domination of innovative economy so at modeling it is required to formulate some general methodological principle of such allocation. The purpose of the given work – to describe it is formal in enough general view and to consider its new appendices, on an example of optimizations of strategy of mining operations. On the constructed mathematical models the basic ways of optimizations and acceptance of scientifically well-founded decisions are revealed.

1.    Methodology and result

Let’s characterize a condition of some biological population number: N ¹ , N ² , N ³

• 1    group (younger) – from 0 till 1 year.

• 2    group (average) – from 1 till 2 years.

• 3    group (senior) – from 2 years also is more senior.

The characteristic of groups:

• 1    and 2 groups – sale of live cattle (breeding) is possible,

• 2    and 3 groups – manufacture of meat and other meat products, 3 groups– manufacture of dairy production.

The account to an age and sexual sign is conducted.

Let’s enter factor, considering a forage reserve, it be artificial the prepared for ages, or grazing (pastures).

The basic dynamics of number of biological population is described by system of the equations:

N & ¹ = ( ст 3 to N 3 ) “ - k 1 N ¹ - U 1

N & ² = y 2 N ¹ - k 2 N ² - U 2

N ³ = y 3 N ² - k 3 N ³ - U 3

j k |^ q 1 U 1 + q 2 U 2 + q₃U ₃ + ( qpwN ³ ) - ( p 1 N ¹ + p 2 N ² + p ₃ N ³ )^1 ^ max (1) ц - Quantity of dairy production for 1 year.

• o ₃ - Fruitfulness factor.

• w – Factor of cows in herd.

• q 1    – Sale price for population unit on the first group.

• q 2 q 3 – Average cost production meat - –n the second and third group. p ₁ , p ₂ , p ₃ – Maintenance cost in corresponding groups for a livestock. q – The price for dairy production.

• Y 2 , Y ₃ — Indicators of intensity of transition from one group in the following.

k , k , k – Natural decrease of population.

It is required to define a policy of conducting such economy on the set time interval, [ t n , t_k ] i.e. the functions u 1 ( t ) >  0 , u ² ( t ) >  0 , u ³ ( t ) >  0 , providing the maximum value functional (1) under following boundary conditions:

N ( t n ) = N i N ( t k ) > N k i = 1 , 2 , 3

Task in view we will investigate a method of multiple maxima. Let's’make function “K” the following structure:

K ( t , N ¹, N ², N ³)

^-ф N Д$ ^{( wN} 3 ) ^- k i ^{N 1 -} u i ] ^{+ ф} N 2 [У 2 ^{N 1} - ^k 2 ^{N 2} - u 2^]

^+ф N ₃ ^[ Y s ^{N 2 - k} 3 ^{N 3 -} u 3 ]

+ [ g 1 U i + g 2 u 2 + g 3 u 3 + g Ц wN ³ - ( P ¹ N ¹ + P 2 N ² + P ³ N 3 )]

+ф ; ( t , N ⁱ, N ², N ³)

Here    p N _i , p N ₂ , p N ₃    - private derivative functions of Krotov

Ф ( t , N \ N ², N ³).

The common decision:

u i : ^-ф N i ⁺ g i ^- 0

^U 2 ^: -^ф _N 2 ^{+ g} 2 ^{- 0}

^U 3 : -^ф _N 3 ^{+ g} 3 ^{- 0}

Looks like

ф ⁽ t , N i, N 2, N 3 ) ^- g i N i + g 2 ^N 2 + g 3 N 3 +ф ^Г

Ф\ - Private derivative of function ^{ф (} t , ^N i, ^N 2 , ^N 3 )

Let's’copy function K taking into account (2)

K ^- N i l ^- g i k ⁺ g 2^2 ⁺ g 3 Y 3 — P i^{] + N} 2^{[ -} g 2 k 2 — P 2^{] + N} 3 lPW — g 3 k 3 — P ₃ ^{] +} 3 ф r '

Let's enter designations:

^Qi ^-- g i k i ⁺ g 2 Y 2 ⁺ g 3 Y 3 ^- P i

^Q 2 ^- g 2 Y 2 ^- P i

Q 3 ^-Ц wg ^- g 3 k 3 ^- P ₃

Depending on a sign Q ₁, Q ₂, Q ₃ – various cases are possible.

Case i Q i >  0, Q 2 <  0, Q 3 >  0

Let's’put ф ( t , N _i , N ₂ , N ₃) = 0

We solve a problem on max functions ф and min functions G which looks like

G ( N^l_n , N k ) - g i N i ( t k ) + g 2 N 2 ( t k ) + ф \.

At the moment t - t k it is visible, from boundary conditions that

G ^ min for N ( t k ) = N k

K ^ max <

u 1 ( t ) = 0 u 2 ( t ) = 0

u 3 ( t ) = 0

Strategy of sale of live cattle and the face, providing the maximum value functional, consists in population "accumulation" 2nd an interval [tntk)(ui (t) - 0, i = 2,3 on this interval) with the subsequent withdrawal at the moment of time t = tk of cattle for a meat-packing plant. Certainly, this conclusion is fair, if the decision N1(t), N2(t), N3(t) corresponding to managements u2(t) = u3 (t) = 0, satisfies to inequalities N (tk) > N‘k, i = 2,3. Otherwise in general there is no the management satisfying to set boundary conditions.

It is easy to see that if restrictions on management look like u ²( t ) >  u 2 , u ³( t ) >  u 3 , the optimum policy should consist in carrying out of a craft with the minimum intensity on all interval of planning, except for a final stage on which intensity of a craft sharply increases. Here u_bⁱ – the bottom borders of management which should be defined with the account both economic and biological factors. In particular, infringement of these borders can cause decrease in number of population to unfairly low level.

Conclusion

The problem of rationing of influence by biological populations on environment consists in the following. At repeated excess of biological population in the limited territory where there is an economic interest, very vulnerable component of the world of the wild nature, untouchable is destroyed. Following ways of development with the assistance of the person are possible, or completely it is possible to destroy this or that resource, or to give the chance to (promote) boundless growth (development), in particular a biological resource where the problem of management and rationing questions become actual. It is required to establish norms on intensity of operation of resources, a biological resource at which observance of norm on quality of water and ground resources will not be broken for long enough period of time.