Review Article Volume 4 Issue 1
Institute of Automation and Control Processes, Russia
Correspondence: Aleksandr I Abakumov, Far Eastern Federal University, Russia, Tel +7423 2310439
Received: January 30, 2018 | Published: February 6, 2018
Citation: Abakumova AI. The dynamic properties in the matrix model of biological system under optimal harvesting. Int Rob Auto J. 2018;4(1):47-49. DOI: 10.15406/iratj.2018.04.00091
The problem of optimal harvest in biological system is investigated. The structure of system is described by matrix model. The properties of stabilization indexes of harvest components are shown. The harvesting regime is stabilized on low level for long time interval of harvesting. These properties are analogous for turnpike theorems in models of economic dynamics.
Keywords: matrix model, biological system, optimal harvesting, turnpike phenomenon
Matrix models are widely used for describing of ecological, economic and social processes. This article has results for biological applications. The matrices are nonnegative usually by describing of abundance dynamics for biological populations or communities.1 For example the abundance dynamics for a population with age structure may be described by model (1) with matrix A as Leslie matrix L.2 The results of O Perron and of G Frobenius for nonnegative matrices are widely known.3,4 A model of abundance dynamics for biological system has form:
{xt+1=Axt,t=1,2,...x1=ˉx≥0, (1)
Here А is a nonnegative irreducible matrix of order n[1, 2, 3], parameter t denotes a discrete time, vector xt∈Rn+ describes an abundance for all groups of the system.
The next result follows for model (1) from Perron-Frobenius Theorem [3,5]: if h is the order of cyclicity of matrix A, then limt→∞1ht+h−1∑τ=tXτ+1rτ=(q⋅ˉx)p . The vector p is positive eigenvector for spectral radius ρ(A) of matrix A, the vector q is positive eigenvector for spectral radius ρ(A*)=ρ(A) of transposed matrix A* . These vectors are connected by condition q⋅p=1 . This result about asymptotic properties in matrix model is very important. We will compare these asymptotical properties of the abundance dynamics of biological systems with the dynamics under optimal harvesting.
We formulate a problem of the optimal harvest in matrix model (1):
{T∑t=1wUtxt→ut∈[0;1]supxt+1=A(I−Ut)xt x1=ˉx (2)
Vector w characterizes the specific income for specific yield. Matrix Ut=diag(ut) contains the harvest vector on main diagonal, 0≤Ut≤I . The time is measured in years. The planning time interval is T. All vectors are nonnegative. The abundance dynamics of biological system is researched in this work. The asymptotic properties for the problem of the optimal harvest have other forms than the properties for model (1).
We study the problem (2) for nonnegative irreducible matrix A. The denotation
yt=(I−Ut)xt (3) will be used. Then next equations are true:
xt+1=Ayt,Ut+1xt+1=xt+1−yt+1=Ayt−yt+1,U1x1=ˉx−y1 Optimization criterion is being transformed to the form: w((ˉx−y1)+T−1∑t=1(Ayt−yt+1))→sup . The equations in model (2) are being transformed to form: 0≤yt≤ˉx and 0≤yt+1≤Ayt for t=1,...,T−1 .
Lemma 1: The problem (2) is equivalent to a next problem:
{Φ=T−1∑t=1w[A−I]yt−wyT→sup0≤yt+1 ≤Ayt ,t=1,...,T−1 0≤yt≤ˉx . (4)
Proof: Every possible solution for problem (2) is transformed to possible solution for problem (4) by formula (3). Inverse formulas have forms:
x1=ˉx,xt+1=Ayt,utj={xtj−yforxtj>0every value from [0,1]forxtj=0,t=1,...T−1 (5)
The optimal solution for problem (2) corresponds to a optimal solution for problem (4).
The following denotation will be used: ψ=w(A−I) .
Lemma 2: The optimal solution for the problem (4) has form:
{ˆy1diag(sgψ)ˉxˆyt+1=diag(sgψt+1)Aˆyt for t=1,2,....,T−2ˆyT=0 (6)
We use here the next denotations:
{φT−1=ψφt=ψ+φ*t+1 for t=1,2,...,T−2 (7)
a+=max{a,0},sga=(0for a<01 for a≥0 . (8)
Proof: The equation ˆyT=0 follows from no negativity of vector w (symbol “^” corresponds to some optimal solution). The denotation Φt=t∑τ=1ψ·yτ is being used. The equation Φ = ΦT−1 is true for optimization functional. We will find the optimal value for variable y with help of induction by decrease of variable t.
Step (T-1). We are getting formulas: Φ = ΦT−1=ΦT−2+ψ⋅yT−1=ΦT−2+φT−1yT−1 for φT−1=ψ The support task {φT−1yT−1→sup0≤yT−1≤AyT−2 has the solution in form: ˆyT−1=diag(sgφT−1)AyT−2 . The general form for this solution is ˆyT−1=ˆyT−1(yT−2) . Next formula φT−1ˆyT−1=φT−1ˆyT−1(yT−2)=φ+T−1AyT−2 is being calculated. The formula Φ=ΦT−2+φ+T−1AyT−2 is true.
Step t. Let formula Φ=ΦT+φ+t+1Ayt be true. Then Φ=ΦT+ψyt+φ+t+1Ayt=Φt−1+φtyt is true for φt=ψ+φ+t+1A The solution of the subproblem {φtyt→sup0≤yt≤Ayt−1 is forming ˆyt=diag(sgφt)Ayt−1 The formula Φ=Φt−1+φ+tAyt−1 follows from previous formulas.
The lemma 2 is proved by induction by decrease of variable t. Corollary. The optimal solution in problem (2) exists and has form:
{ˆx1=ˉx ˆxt+1=A(diag(sgφt))ˆxt,t=1,2,...,T−1ˆuT=e ˆut=¯sgφt .
The denotations ¯sga={1,a<00,a≥0 , e=(1,1,...,1) are used. Proof is based on use of formulas (5).
Theorem. The solution of the problem (2) with use formulas (3) has the time points tjand sets Ij⊂{1,2,...,n} of indexes for j=1,…,k by some k≤n . These points and sets have next properties: 1=t1<t1<...<tk=T and {ˆyti=0for i∈Ij ˆyti>0for i∉Ij and t∈(tj;tj+1) , Ij⊂Ij+1 .
Proof. The property φT−1−φT−2=ψ−(ψ+φ+T−1A)=−φ+T−1A≤0⇒φT−1≤φT−2 follows from formulas (7). We have φt−φt−1=(φ+t+1−φ+t)A for t=2,3,...,T−2 . The inequalities
φT−1≤φT−2≤φ1 (8)
Follow from previous formula.
The results of the theorem follow from the lemma 2 and formula (8). The meaning of this theorem is in the turnpike phenomenon6 for the solution of problem (2). This phenomenon consists in stabilization of the indexes of the system components which are taken in harvest. This stabilization is being discussed in next part.
The variable yti denote the abundance of component i for time moment t after harvesting. The theorem means that the solution for optimal harvesting problem (2) has the index sets Ij of zero components after harvesting. These sets have the inclusion properties Ij⊂Ij+1 . This situation is analogous for turnpike phenomenon in models of economic dynamics.6 But the turnpike properties are specific in this case. These turnpike properties are true for indexes of harvesting components: if the component equals zero after harvest in some time point then this component equal zero in any next time points. The inclusion Ij⊂Ij+1 means that the new Biosystem components are being included in harvest with increase of time. For example this fact means a decrease of average age of individuals in population with age structure.
The first index set I1 denotes the lightest regime of harvesting. The time moments tj for j≥2 go to infinity if interval for harvesting went to infinity T→∞ . The lightest regime of harvesting dominates in this case. The long time planning interval is preferable to efficient use of biological resources.
My research project was partially or fully sponsored by Program of fundamental scientific research “Far Eastern” of Far Eastern Branch of the Russian Academy of Sciences (years 2015-2017) with grant number 15-I-4-006_o.
None.
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