Properties of Admissible Set of an Optimal Non-destructive System Exploitation Problem in Some General Formalization

Abstract

We study the properties of admissible strategies for the problem of using of a renewable resource system that does not lead to its destruction, in some general formulation. It is assumed that the system evolution is described by some iterative process. The conditions for the stable existence of this system, originally formulated in terms of the asymptotic properties of this iterative process, reduce to the existence of admissible controls for some mathematical programming problem. We analyzed the properties of its admissible set in terms of the dominant eigenvalues of some positively homogeneous maps. We obtained the conditions for the existence of optimal controls under the in uence of which the system can stably exist for an inde nitely long time. Introduced by author's generalization of the classical concept of the irreducibility of a non-linear map, the concept of local irreducibility, is essentially used. The problem under consideration has important applications in the tasks of rational exploitation of renewable natural resources, in particular, in the tasks of managing ecological populations.

Keywords

• Concave programming
• Irreducible map
• Positive equilib- rium
• Rational exploitation of natural resources

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One of a vital problems that require the use of mathematical methods is a problem of a rational use of renewable resources. Particularly acute in practice is a problem of rational non-destructive ecosystem exploitation. This problem is characterized by an abundance of speci c approaches for speci c natural systems (see the review in [4]). In the matrix formulation, the approach to optimal exploitation of a population in a stationary state has prevailed for a long time. This approach uses as a structure of the population a stable distribution | the structure of an eigenvector corresponding to the dominant eigenvalue of the Copyright c by the paper's authors. Copying permitted for private and academic purposes.

In: S. Belim et al. (eds.): OPTA-SCL 2018, Omsk, Russia, published at http://ceur-ws.org projection matrix (that must be greater than unity). If at some point in time the structure of the population coincides with a stable distribution, then it is possible to remove the surplus without damage to the population, returning the population to the previous state. This corresponds to the withdrawal of the same proportion of all its constituent groups.

One of the rst theoretical works in this direction was [2], where the concept of sustainable yield was de ned, and the solvability of the problem of nding the maximum level of permissible exploitation was proved. In work [1] some concepts natural for the exploitation of ecosystems were formalized and the corresponding statements were rigorously proved. The rst successful attempts to generalize these results to the case of a nonlinear density dependence are related to papers [11, 3] that were considered the dependence on density only for the rst age class.

In [5] was considered the model with dependence on density for all age classes. We are developing here a general approach to this problem that considers the ecosystem as a discrete system, with its representation in the form of an iterative process

xt+1 = F (xt); t = 0; 1; 2; : : : ( 1 ) where xt means the state of the system at time t = 0; 1; 2; : : :, the step operator F transforms the system from the previous state to the following. With this approach, one can naturally formalize the inadmissibility of excessive load on the dynamic system condition.

We will use the following notation: Rq+ denotes the non-negative orthant of Rq; x y means y x 2 Rq+; x < y means y x 2 int Rq+; int M | the interior of the set M ; x y means x y and x 6= y; Z | the set of integers; m; n = fi 2 Z j m i ng. Vector x = (x1; x2; : : : ; xq) can be written in the short-hand form as x = (xi). The iteratives of the map F are denoted by F t(x) (t = 1; 2; : : :), F 0(x) x.

A controlled system is modelled by iterative process

xt+1 = Fu(xt); t = 0; 1; 2; : : : ; ( 2 ) where x = (xi), u = (ui), Fu(x) = (F (x) u))+, a+ = maxfa; 0g, x+ = (xi+). We will denote by fxt(x0; u)gt+=10 , xt(x0; u) = (xit(x0; u)) the realization of this process starting from the initial vector x0. In [7] we formulated an optimization problem for dynamical system ( 1 ): determine c~ = maxfc(u) j u 2 U g; ( 3 ) where c(u) | the total e ect of using of system elements in the quantities u1; u2; : : : ; uq, the set U is the closure of the set

q

U = fu 2 R+ j X0(u) 6= ?g of controls. We will be more interested in the properties of the admissible set U , so for simplicity we shall assume that c(u) is linear nonnegative on Rq+ function.

The control vector u is considered as admissible if there is at least one initial

state x0, for which all units of the system stably exist for an inde nitely long time:

q

X0(u) = fx0 2 R+ j intf xit(x0; u) > 0 (8i 2 1; q)g

Thus, the size of each unit of the system should not decrease with time to

zero. As can be seen from this de nition, the set U formalizes the requirement of the indestructability of the managed object. the AmanpecFes,sai.rey.,ccoonnddititioionnfoNrF+U6 =6=??. Iins tthheisecxaissteenthcee soeftaUpocsointitvaeinsxaetdlepaositntthoef trivial control: 0 2 U .

Unfortunately, the set U is not always closed; thus, in problem (3), its closure is considered as the admissible set.

Denote by u~ and Ue the optimal vector and optimal set of problem (3); and

by Nu and Nu+ the sets of nonzero xed points and positive xed points of Fu, respectively. It is assumed that the original model without control ( 1 ) has the trivial equilibrium state: F (0) = 0.

We are interested in the positive realizations of the iterative process (2) only,

when xt(x0; u) > 0 (8t = 1; 2; : : :). In this case Fu(x) = F (x) u > 0 for all x = xt(x0; u). Note that if x 2 Nu+, then Fu(x) > 0 and F (x) = x + u also must be ful lled.

The main requirement imposed on the class of maps under consideration is

a concavity or its weakening, a subhomogeneity [6]. The de nitions of subhomogeneous and superhomogeneous maps, as well as other standard concepts used below, are given, for example, in [8]. The subhomogeneity of a map make it possible to depict adequately a so-called saturation e ect observed in many natural systems with limited resources.

In what follows, the concept of irreducibility (indecomposability) of a map

is essentially used. Along with the classical concept of irreducibility [10], we will also use its local analogues [12], such as a notion of irreducibility of a map at zero. We give a classical de nition of the irreducibility of a map in a somewhat di erent way, using the following sets:

I+(x; y) = fj 2 1; q j xj > yj g;

I0(x; y) = fj 2 1; q j xj = yj g; I+(x) = fi 2 1; q j xi > 0g;

I0(x) = fi 2 1; q j xi = 0g: A map F is said to be reducible at a point y 2 Rq+, if 9x 2 Rq+ : x y;

I0(x; y) 6= ?;

I0(x; y)

I0(F (x); F (y)): ( 4 )

A map F is said to be irreducible at point y, if it is not reducible at point y.

A map reducible (or irreducible) at every point of set M is said to be reducible

(or irreducible, respectively) on the set M .

We consider separately the case of irreducibility at the point y = 0. A map q q F 2 fR+ 7! R+g is said to be reducible at point y = 0 (reducible at zero), if 9x 2 Rq+ : x

We note that the concept of reducibility at zero makes sense only for maps

satisfying condition F (0) 6> 0. q

The global irreducibility of a map means its irreducibility at all points R+

and, in particular, irreducibility at zero.

In a research of asymptotic properties of iterative processes a primitivity

property of a map [10] is usually used. Although irreducibility at zero is a weaker property in comparison with primitivity at zero, it also guarantees the positivity of nonzero points of a map. Indeed, suppose, by way of contradiction, that I0(x) 6= ?. Then from fi(x) = xi (8i 2 I0(x)) we get I0(x) I0(F (x)), which conradicts, by ( 5 ), the irreducibility of F at zero. Therefore I0(x) = ? and x > 0.

The local irreducibility properties for monotone increasing subhomogeneous maps are considered in [8, 9].

We re ne our assumptions about properties of dynamical system (1). The

step operator F is assumed to be nonnegative and concave on Rq+ (and hence it is monotone increasing on Rq+). It is also natural to assume that the condition for the absence of identically zero coordinates of F is satis ed: ( 6 ) It is important that this property for monotone increasing subhomogeneous map F implies that F (int Rq+) int Rq+. A su cient condition for ( 6 ) is the irreducibility of F at zero. Since irreducibility at zero also guarantees the positiveness of non-zero xed points, it is convenient to assume from now on that the map F is irreducible at zero. 2

Some Preliminary Results For the subhomogeneous map F we can de ne the following positively homogeneous maps:

F0(x) = lim !+0 1F ( x);

F1(x) = lim !+1 1F ( x);

It is easy to see that

De ne the following sets:

F1(x)

F (x)

F0(x) (8x 2 Rq+): PF+ = fx > 0 j F (x) > xg;

QF+ = fx > 0 j F (x) < xg: It is shown in [13] that for a subhomogeneous, monotone increasing and irreq q ducible at zero map F 2 fR+ 7! R+g the following properties are valid: (F0) > 1 , PF+ 6= ?; (F1) < 1 , QF+ 6= ?: ( 7 ) ( 8 ) ( 9 ) Likewise, for the superhomogeneous, monotone increasing and irreducible at zero map F the following properties are valid: (F0) < 1 , QF+ 6= ?; (F1) > 1 , PF+ 6= ?:

We introduce the set and quantities

MF (x) = f > 0 j F ( x =

F (x)g;

F (x) = inf MF (x); F (x) = sup MF (x); where x is an arbitrary nonzero vector. Note that the set MF (x) always contains unity as an element.

Existence conditions of positive xed points for subhomogeneous maps on q cone R+ were studied in [13]. It follows from ( 8 ) due to [10, Theorem 10.3] that the condition (F1) 1 (F0) is necessary for existence of a positive xed point for monotone increasing, subhomogeneous and irreducible at zero map. A similar condition with strict inequalities:

(F1) < 1 < (F0) is su cient for existence of a positive xed point. Su cient conditions with (F1) = 1 or (F0) = 1 require the use of the global irreducibility property.

With some work it can be shown that if F is monotone increasing, subho

mogeneous, and condition ( 14 ) is satis ed, then NF+ is bounded, convex and contains a largest element xF . Moreover, a concave globally irreducible map satisfying condition ( 14 ) has a unique positive xed point [13, Corollary 2.2.1].

If Nu+ = ? for all u > 0, then it follows from condition NF+ 6= ? that U = f g

0 . This case is uninteresting to us, therefore, in addition to condition NF+ 6= ?, it is necessary to assume that Nu+ 6= ? for some u > 0. In this case F (x) = x + u > x (8x 2 Nu+), and by ( 10 ), we have (F0) > 1. For this reason, the condition ( 14 ) is assumed everywhere below. Then the set U is also convex, contains some positive vector and with each positive vector u also contains the segment [0; u]; those U is so-called downward set. Besides the following equalities hold:

q U = fu 2 R+ j Nu+ 6= ?g; q

U = fu 2 R+ j Nu 6= ?g;

Finally, it was shown in [7] that under above assumptions the problem of mathematical programming maxfc(u) j x = F (x) u; x 0; u 0g is solvable in the sense that c~ < +1 and this value is attained on certain admissible vectors u~, x~. Moreover, c~, u~ is a solution of problem ( 3 ) if and only if c~, u~, x~ is a solution of problem (16).

We now turn to properties of the admissible set for this problem. Proceeding from its basic interpretation ( 10 ), we will be interested rst of all in properties and conditions for existence of the positive equilibria x~ for the control u~. ( 11 ) ( 12 ) ( 13 ) ( 14 ) (15) (16)

Some Properties of the Admissible Set of Problem ( 3 ) Note that the set Nu for u 2 U contains the largest element xu and the map x(u) : u ! xu is monotone decreasing on U . It can also be shown that x(u) inherits the concavity of the map F .

We introduce the set

q D = fu 2 R+ j Nu 6= ?; Nv = ? (8v > u)g (17) that forms part of a boundary of U . The set D is of interest in connection with the fact that it contains all potentially optimal vectors of the problem ( 3 ): Ue D. We divide this set into two disjoint parts by the criterion of the presence or absence of common points with U :

From equalities (15) we obtain the following representation of D00:

D0 = D n D00;

D00 = U n U: q

D00 = fu 2 R+ j Nu 6= ?; Nu+ = ?g: We note that, by de nition of D00, for elements of Nu with u 2 D00 the following property is true:

xu 2= int Rq+ (8u 2 D00):

Directly from (17){(19) the following representation follows:

D0 = fu j Nu+ 6= ?; Nv = ? (8v > u)g: The representations (19), (21) can be clari ed. Indeed, it follows from (21), in view of (19), that D0 U . Since D0 D, we get D0 D \ U . On the other hand, D \ U D0, because u 2 U together with the second equality in (18) means u 2= D00. Hence, taking into account u 2 D and the rst equality in (18), we obtain u 2 D0. Therefore, D0 = D \ U .

Further, from the second equality in (18) it follows that D n U D00. On the

other hand, D00 D n U , since D00 D and D00 \ U = ?. Therefore D00 = D n U .

Thus, we have the following equalities:

D0 = D \ U;

D00 = D n U: These equalities show that the positive optimal vector x~ for the optimal control u~ of the problem ( 3 ) exist if and only if u~ 2 D0. Using (22), we can re ne that U n D = U n D0.

The set U n D does not a fortiori contain the optimal vectors of the prob

lem ( 3 ). We give without proof the following assertion characterizing this set.

q Lemma 1. If F is concave on R+, irreducible at zero and the condition ( 14 ) is satis ed, then u 2 U n D if and only if there exists v > u belonging to the same set: v 2 U n D; or equivalently, if and only if there exists x > 0 such that Fu(x) > x. (18) (19) (20) (21) (22) Let us de ne the maps y 2 fR+ 7! Rq+g, y 2 f[0; y] 7! Rq+g, where y 2 Nu, q y(x) = ( iy(x)), y(x) = ( yi(x)), with the following equalities: y(x) = F (x + y)

F (x); y(x) = F (y)

F (y x): (23) Obviously, xu xF ; therefore we can assume that y, which plays the role of a parameter here, varies in [0; xF ].

It is easy to see that the maps y; y inherit the monotonicity of the map F , iass awlseoll caosntchaveeporenseRnq+ce, aonfda tthrievimalapxeyd ipsocinotnvxex=o0n. [F0u; ryt]h.eTrmheorreef,orteh,efomratphesye maps there exist positively homogeneous (of the rst degree) maps ( y)0; ( y)0 de ned for F in ( 7 ). For their components the following equalities hold: ( iy)0(x) = fi0(y; x); ( yi)0(y; x) = fi0(y; x) (8i 2 1; q); where fi0(y; x) is one-sided directional derivative of the function fi(x) at the point y in the direction x.

The maps y, y (and, hence, their dominant eigenvalues) are related by the followings inequalities: (24) (25) y(x)

y(x) (8x 2 [0; y]); (( y)0)

(( y)0): ( y)0(x) = ( y)0(x) = F 0(y)x; (( y)0) = (( y)0) = (F 0(y)): If F is di erentiable at point y and F 0 is its derivative, then the following equations are valid:

The following relations between the sets of xed points for the maps Fu, y; y and the sets (9) for these maps follow directly from their de nitions.

q Lemma 2. Suppose that F is concave on R+, irreducible at zero and the condition ( 14 ) is satis ed. Then the following properties are valid: ( 1 ) x 2 P +y ; y 2 Nu ) x + y 2 PF+u ; ( 2 ) x 2 Q+y ; y 2 Nu; x < y ) y x 2 PF+u ; ( 3 ) x 2 N y ; y 2 Nu ) x + y 2 Nu; ( 4 ) x 2 N y ; y 2 Nu; x y ) y x 2 Nu; ( 5 ) x 2 Nu; y 2 Nu; y x ) x y 2 N y : ( 6 ) x 2 Nu; y 2 Nu; y x ) x y 2 N +x :

We show that y and y inherit also the global irreducibility of the map F .

Lemma 3. Suppose that F is concave on Rq+ and the condition ( 14 ) is satised. If the map F is globally irreducible, then the map y (y 2 Rq+) is globally irreducible, and the map y (y 2 int Rq+) is irreducible on [0; y). Proof. Let F be globally irreducible and suppose for contradiction that y is q not globally irreducible for some y 2 R+. By ( 4 ) this means that there exist x; x0 such that x0 x, I0(x; x0) 6= ? and i(x; y) = i(x0; y) for i 2 I0(x; x0). Then for z = x + y, z0 = x0 + y we have: z0 z, I0(z; z0) = I0(x; x0) 6= ? and fi(z) = fi(z0) for all i 2 I0(z; z0). Hence F is not globally irreducible, which contradicts our assumptions. Thus, y is globally irreducible. The proof of the irreducibility for y is carried out analogously. The proof is complete.

We proceed to describe the set D by means of the maps (23).

q Theorem 1. Suppose that F is concave on R+, irreducible at zero and the condition ( 14 ) is satis ed. If u 2 U , then u 2= D if and only if (( xu )0) < 1. Proof. Necessity. Using (15) and Lemma 1 for u 2 U nD, we obtain that Nv+ 6= ? for some v > u. Then, denoting xu xv > 0 by x, we get: xu (x) = F (xu) F (xu x) = F (xu) F (xv) = (xu+u) (xv +v) = x+(u v) < x, i. e. xu (x) < x. It then follows, by ( 11 ) due to convexity (and, hence, superhomogenity) of the map xu , that (( xu )0) < 1.

Su ciency. If (( y)0) < 1, then, by (11), there exists x, such that 0 < x < y

and x 2 Q+y . Hence, due to the assertion ( 2 ) of Lemma 2, we obtain y x 2 PF+u , that is, Fu(y x) > y x. But this means, by Lemma 1, that u 2 U n D, as required.

Now we can obtain a characterization of the set D in terms of the maps (23).

q Corollary 1. Suppose that F is concave on R+, irreducible at zero and the condition ( 14 ) is satis ed. If u 2 U , then u 2 D if and only if (( xu )0) 1.

The following statement gives a su cient condition for an admissible vector not to belong to the set D.

q Theorem 2. Suppose that F is concave on R+, globally irreducible and the condition ( 14 ) is satis ed. If u 2 U and (( x)0) > 1 for some x 2 Nu, then u 2 U n D.

Proof. By ( 10 ) and Lemma 3, there exists x > 0 such that x(x) > x. It then follows, since assertion ( 1 ) of Lemma 2, that Fu(x0) > x0 for x0 = y + x. But this means, by Lemma 1, that u 2 U n D. The proof is complete. q Corollary 2. Suppose that F is concave on R+, globally irreducible and the condition ( 14 ) is satis ed. If u 2 D and y 2 Nu then (( y)0) 1.

Combining Theorem 1, Corollary 1 and Corollary 2, we obtain the following characteristic of U for the di erentiable map F .

q Corollary 3. Suppose that F is concave, di erentiable on R+, and the condition ( 14 ) is satis ed. If the map F is also irreducible at zero, then If, in addition, the map F is globally irreducible, then (F 0(xu)) < 1 (8u 2 U n D): (F 0(xu)) = 1 (8u 2 U \ D): (26) Proof. If u 2 U n D, then the inequality (26) follows from Theorem 1 due to (25). If u 2 U \ D, then, by Corollary 1, we obtain (F 0(xu)) 1. On the other hand, according to Corollary 2, we have (F 0(xu)) 1, so (F 0(xu)) = 1. The proof is complete.

Using the property 5 of Lemma 2 and Lemma 3, we obtain the following im

portant property about the impossibility of a partial coincidence for coordinates of xed points of Fu in the case of the globally irreducible map F : 8x; y 2 Nu : x y ) x < y: (28) We will give now one more characteristic feature of the elements of Nu for u 2 U n D. The following assertion supplements the conclusion of Theorem 2. q Theorem 3. Suppose that F is concave on R+, global irreducible and the condition ( 14 ) is satis ed. Then the following properties are valid: (( xu )0) < 1;

(( x)0) > 1 (8 u 2 U n D; x 2 Nu n fxug): Proof. The rst inequality follows from Theorem 1 and inequalities (24). Further, if x 2 Nu n fxug, then, by assertion 5 of Lemma 2, x0 = xu x is the xed point of x that is positive by (28). Hence we obtain (( x)0) 1 [10, Theorem 10.3].

Suppose, by way of contradiction, that (( x)0) = 1. Then due to [13, Theorem 2.2.11], we get (0; 1) M x (x0) (see ( 12 ){( 13 )). We show that the condition

2 M x (x0) is satis ed if and only if the following equality is true:

F (( 1 )x + (x + x0)) = ( 1 )F (x) +

F (x + x0) (8 2 (0; 1)): (29) Indeed, we have: 2 M x (x0) , x( x0) = x(x0) , F ( x0 + x) F (x) = (F (x0 + x) F (x)) , F ( x0 + x) = ( 1 )F (x) + F (x + x0). Since x0 + x = ( 1 )x + (x + x0), we obtain the equality (29). Taking into account that x + x0 = xu, we obtain from here the equality

F (( 1 )x + xu) = ( 1 )F (x) +

F (xu) (8 2 (0; 1)): (30) By assertion 6 of Lemma 2, x0 is a xed point of xu too. We show now that, moreover, if (( x)0) = 1, then all points x0 ( 2 (0; 1]) are also xed points of the map xu .

Indeed, since xu x0 = ( 1 )xu + x, we have xu ( x0) = F (xu) F (xu x0) = F (xu) F (( 1 )xu + x): The equality (30) holds for all 2 (0; 1), so we get: xu ( x0) = F (xu) ( 1 )F (xu) F (x) = (xu + u) (x + u) = x0; so that really xu ( x0) = x0 (8 2 (0; 1]): These equalities mean that ( xu )0(x0) = x0, thus (( xu )0) 1 [10, Theorem 10.3]. But then it follows, by Corollary 1, that u 2 D. This contradiction shows that (( x)0) > 1. The proof is complete.

In conclusion, we give several assertions about a cardinality of set Nu with

u 2 D. If xu is not unique in Nu, then it follows from (28) that xu > xu (8xu 2 Nu n fxug). In this case, we can de ne the set

Lu = cofxu; xug; where co M is the convex hull of a set M . We note that, as follows from the assertion 5 of Lemma 2 and from Theorem 3, for u 2 U n D the set Nu cannot contain the entire segment Lu, along with points xu; xu. But for the D0 the situation is di erent. The following assertion shows that Nu with u 2 D is either a singleton or an in nite set. q Theorem 4. Suppose that F is concave on R+, global irreducible and the condition ( 14 ) is satis ed. Then the following property holds: u 2 D; xu 2 Nu n fxug ) Nu

Lu: Proof. Due to the assertion 5 of Lemma 2, the map xu has a non-zero xed point x = xu xu. The map xu is global irreducible (see Lemma 3), therefore, this xed point is positive. Due to ( 8 ), we get: ( xu )0(x) xu (x) = x, hence (( xu )0) 1 [10, Theorem 10.3]. But, the Corollary 2 gives the opposite inequality, so that (( xu )0) = 1. In this case, the equation xu = 0 must be ful lled, where xu is de ned by ( 13 ) (see [13, Theorem 2.2.11]). This means that xu ( x) = xu (x) (8 2 [0; 1]). Therefore, we obtain for the set of positive xed points of xu that N +xu f x j 2 [0; 1]g. By assertion 3 of Lemma 2 this means that Nu fxu + x j 2 [0; 1]g = fxu + (xu xu) j 2 [0; 1]g = Lu.

The proof is complete.

Taking into account properties (20), (28), we obtain from the Theorem 4 the following assertion.

q Corollary 4. Suppose that F is concave on R+, global irreducible and the condition ( 14 ) is satis ed. Then the following properties hold: jNuj 2 f1; +1g (8u 2 D0); jNuj = 1 (8u 2 D00): (31) If F is also strictly concave on [0; xF ], then jNuj = 1 (8u 2 D).

In conclusion, let us give an example showing the essentiality of the requirement of global irreducibility in the above statements, which used this assumption. This example uses a generalization of so-called Leslie's model [14] of the following form:

(t+1) = fi(at); xi;1 xi(;tj++11) =

(t) i;j xi;j (i 2 1; m; j 2 1; n 1); (32) where at = Pim=1 Pjn=1 i;j xi(;tj), i;j > 0, i;j 0, xi(;tj) 0 (8i 2 1; m; j 2 1; n). The functions fi(a) (i 2 1; m) will be assumed to be nonnegative and concave on R+. The step operator F (x) = (fi;j (x)) of this model has the following components: fi;1(x) = fi(a(x)); fi;j+1(x) = i;j xi;j (i 2 1; m; j 2 1; n 1); (33) where a(x) = Pim=1 Pjn=1 i;j xi;j . Thus, the map F inherits a concavity and monotonicity of functions fi(a).

The admissible set U in this case, in addition to the requirement of non

negativity of variables, is given by the following constraints: xi;1 = fi(a(x)) ui;1; xi;j+1 = i;jxi;j ui;j+1 (i 2 1; m; j 2 1; n 1); (34) Let us introduce the following notation:

m (a) = X (i)fi(a); q(u) = (q; u); i=1 (a) = (a) a;

(35) n where ( ; ) denotes the scalar product, qj(i) = P k=j

k 1 i;k Q `=j s Q a` = 1 for i > k), (i) = q1(i), q = (q( 1 ); q( 2 ); : : : ; q(m)) , q(i) = (q1(i); q2(i); : : : ; qn(i)) `=r (i 2 1; m; j 2 1; n) (k; `; r; s 2 Z).

Multiplying equalities (34) by i;j and summing, we obtain for a = a(x) the equation i;` (by convention q(u) = (a): (36) Thus, if xu 2 Nu, then (36) holds, which for a given u can be regarded as an equation for a. This equation, due to the concavity of (a), can have no more than two solutions for (a) 6= = maxa (a).

Example 1. Consider the system of constraints xi;1 = fi(a)

1 ui;1; xi;2 = 2 xi;1 ui;2; a = x1;1 + x1;2 + x2;1 + x2;2; xi;j 0; ui;j 0 (i; j = 1; 2); where f1(a) = 6a(1 + a) 1, f2(a) = minfa; 1g are non-negative monotone increasing concave functions of a nonnegative argument. We nd from (35): 3 3 9a (a) = 2 f1(a) + 2 f2(a) = 1 + a + 3

minfa; 1g; 2 3 3 9a 3 q(u) = 2 u1;1 + u1;2 + 2 u2;1 + u2;2; (a) = 1 + a + 2 minfa; 1g a:

As shown in [14], condition (14) for the model (32) is equivalent to condition

0(+1) < 1 < 0(0): It is easy to see that in our particular case this condition is satis ed. Next, here F 2 fR+ 7! R+g has the following form: F (x) = (f1(a(x); 21 x1;1; f2(a(x)); 12 x2;1).

4 4 This map clearly satis es all assumptions of previous propositions, with the exception of the global irreducibility. The function (a) reaches its maximum value = 11=2 for a = 2.

We take for illustration u = (16=5; 0; 1=10; 0) and show that 2 D00. Calculate q(u) = 27=5 for this vector. Solving the equation (a) = 27=5, we obtain solutions a1 = 3=2, a2 = 13=5. Further, we nd x(a1; u) = (2=5; 1=5; 9=10; 0), x(a2; u) = (17=15; 17=30; 9=10; 0); so that Nu = fx(a1; u); x(a2; u)g. Therefore, xu = x(a2; u), and from (19) it follows that u 2 D00. Thus, we get jNuj = 2 for u 2 D00, so that the second part of the conclusion (31) of Corollary 4 is not satis ed. In addition, we see that the property (28) is also violated for these vectors.

The matrix F 0(x) has the form of a generalized Leslie's matrix:

2f10 (a) f10 (a) f10 (a) f10 (a)3 L(a) = 664f120=(a2) f20 (a) f20 (a) f20 (a)775 ;

0 0 0 0 0 1=2 0 where a = a(x) (see (33)). To compare the dominant eigenvalue of this matrix with unity, it is not necessary to nd this value, because the followig equation is valid [14]: sgn( (L(a)) 1) = sgn( 0(a) 1); where sgn(x) = jxj=x for x 6= 0, and sgn(0) = 0.

For x = x(a1; u) we obtain (F 0(x)) = (L(a1)) > 1 as (a1) > 1. We see that the conclusions of Theorem 2 and of Corollary 2 for the chosen vector u 2 D also do not hold.

Thus, we have demonstrated that the requirement of the global irreducibility in the above statements is essential.

4

Conclusion In this paper we investigated the properties of the admissible controls of the problem ( 3 ) that can be optimal with appropriate presetting of the objective function. In the context of the interpretation of this problem, from which we proceeded, the question of presence of positive equilibrium x for the iterative process ( 2 ) with given admissible control u is essential. We showed that such controls must be contained in the set D0 de ned by (18). In the case of the di erentiability of the map F , these controls are distinguished from the others, by Corollary 3, with simple characteristic property (27). It turns out that the dominant eigenvalue at the equilibrium point xu should equal unity. In addition, as Corollary 4 shows, only for u 2 D0 the set Nu can contain an in nite number of elements.

The question whether in our assumptions this set is non-empty remains gen

erally open. But in some cases, for some known (and su ciently general) mathematical models used in the modeling of biological communities, it is sometimes possible to establish that D0 6= ?. Preliminary studies show that this takes place for the above generalization of Leslie's model, as was the case in Example 1. It can be shown that in the assumption of concavity of all functions fi(x), the set D0 in this model is always nonempty. Particularly important and, to a degree, unexpected is the fact that the set D0 here is part of some hyperplane. Other key properties of the admissible set of problem ( 3 ) for this and other models call for further investigations.

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