ILL-POSED PROBLEM OF RECONSTRUCTION OF THE POPULATION SIZE IN THE HUTCHINSON–WRIGHT EQUATION

We consider an ill-posed problem of reconstruction of the population size in the Hutchinson–Wright Equation. Regularized solutions were constructed on the ﬁnite interval of the negative half-line.


Statement of the Problem
Consider the population model represented by the Hutchinson-Wright equation 1) where N : R → R + = (0, +∞) is the population size, r is the intrinsic growth rate (the birth rate), k is the carrying capacity, h is the delay parameter (the breeding age). The work [1] is devoted to the investigation of various biological factors affecting changes in the number of population. In the analysis of regularities of the change of population the results of observations of the size of biological populations [2] are used. Full-scale observation of changes of population size requires significant material and labor costs. Mathematical modeling can facilitate solution of the problem of studying the dynamics of the quantitative change of population size [3].
The Hutchinson-Wright equation describes a mathematical model of the single-species biocenosis, when the influence of predators is slight, the habitat is homogenous, migration processes do not have a significant impact on the change of population size and the quantity of available food is restored regularly up to a certain level.
The works [3,4] are devoted to the investigation of various qualitative problems of the change of population size for the Hutchinson's equation. The computer simulation of such changes was held in [5]. The motivation of these works is joint with the forecast of changes of population size. The correctness of the initial problem for the Hutchinson's equation helps to solve this problem successfully. The main qualitative result is that the population size tends to constant when time increase or periodic oscillations take place. It is depend on the value of parameter h. If the population is on the stage of a transient of changes of the population size, the inverse problem of determining its size in the previous periods of time is of interest. From the mathematical point of view this problem is more difficult, because it belongs to a class of ill-posed problems [6].
In the present paper the ill-posed problem of reconstruction of the population size in the Hutchinson-Wright equation is solved.
It is supposed that an information about the population size is known on the time interval [t 0 − h, t 0 ]. In the sequel, without loss the generality we assume that t 0 = 0. The population size on the interval [−h, 0] is defined by a positive function ϕ which belongs to a separable Hilbert space When reconstructing the number of population we use the method of steps in the direction of decreasing time. Then for the finding functions x m (ϑ) = N (mh + ϑ), ϑ ∈ [−h, 0], m ≤ −1, we have a system of equations where the operator U : H → H is determined by the formula Thus, the reconstruction of population size is associated with solving the ill-posed problem

Determining System of Equations for Finding the Values of the Regularizing Operator
We use the regularization method of A.N. Tikhonov to solve the ill-posed problem. So we choose a stabilizing functional of the form where G, P , and Q are positive numbers, x is the derivative of the function x. One needs to find an element x α ∈ W 1 2 [−h, 0] minimizing the smoothing functional for a fixed positive value of the regularization parameter α. We obtain the necessary condition for a minimum of the functional [7, p. 113] when finding the Gâteaux derivative of the smoothing functional As a result we have Here the Gâteaux derivative of the operator U in the point x determined by the formula The adjoint operator U * x (x) given by the formulas By using the definition of inner product of the space H, we reduce the necessary condition for the minimum of the smoothing functional to the form and must be valid for any y ∈ H. A possible minimizing element x α satisfies the system of equations By introducing auxiliary functions ψ and χ by the formulas and by using the representations of the operators U and U * x (x), one can replace system of equations for the minimizing element by an equivalent boundary value problem for ordinary differential equations. Assertion 1. The possible minimizing element x α is a component of the solution of the following system of ordinary differential equations with the boundary conditions As it appears from the definition of the function χ it satisfies to the differential equation Once again, from the definition of the function ψ we have that it satisfies to the differential equation with the boundary condition By using the auxiliary functions we rewrite the second equation of the system (2.2) By taking into account the value of the auxiliary function ψ(−h), we reduce the last equation to the form Let us eliminate the auxiliary variables ψ and χ from the system of equations (2.3) and the bounadry conditions (2.4). When finding a solution of this problem we impose additional conditions on the initial function and solution of the boundary value problem, such as ϕ ∈ W 1 2 [−h, 0] and max ϑ∈[−h,0] |χ(ϑ) − ϕ(ϑ)| is a small value. The last restriction is compatible with a statement of the problem of regularization and allows to find the following from the second equation of the system (2.3), as well as guarantee the fulfillment of the condition By calculating a derivative of the last expression and by substituting it into the third equation of the system (2.3), we obtain When differentiating the first equation of the system (2.3) twice, we find By eliminating ψ from equation (2.6) and by introducing variables x , we obtain the system of differential equations We pass the above-introduced new variables in the boundary conditions (2.4). So we have By introducing the vector X = x j 4 1 , we rewrite the system (2.7) in the vector form Let us make a replacement of variables in the system (2.9): 4 1 is a matrix reducing A (ϑ) to the Jordan form, λ(ϑ) = (rϕ(ϑ)/(k √ P )) 1/2 , e 1 =ē, e 2 = e, e 3 = −ē, e 4 = −e, and e = (1 + i)/ √ 2. Then we obtain the system . Then the components of the solution of the system (2.11) are determined by the asymptotic formulas where D j are arbitrary constants belonging to C n , ϑ 1 = ϑ 2 = 0, and ϑ 3 = ϑ 4 = −h. By using the method of variation of constants for the nonlinear system (2.11), we have By integrating the last equalities, we obtain where D j are new constants. By substituting the last expression into (2.13), we find where α, y(s, α, D(s)))) ds.
By applying the integration by parts formula to I1 , we have By substituting the last formula into (2.14), we obtain a system of nonlinear integral equationŝ are new arbitrary constants. By using the principle of contraction mapping [8], we will show that the system of equations |Φ j 2 (s, α,ŷ(s))/λ(s)|. By using the Lipschitz condition Consequently, the operator A is the contracting operator on Ω. Then by using the method of successive approximations [8] and by using the equality Af − f ∞ = O(α 1/4 ), we find

Dependence of the Regularization Parameter on the Admissible Error
Since U x α = χ α , it follows that the discrepancy equation (U x − ϕ, U x − ϕ) = δ 2 acquires the form By using the variables x 2 and x 4 in (2.5), we have By taking into account the formulas (2.16), we find As a result the discrepancy equation acquires the form It can be shown, that the sufficient condition [8, p. 125] for the minimum of the functional M α [ϕ, ·] is fulfilled. By substituting (3.2) into (3.4) we find the asymptotic formulas for the values of the regularized operator on the set D.
Let us introduce an operator R 1 : W 2 ∞ → H by the formulas and an operator R 2 : W 2 ∞ → H by the formulas (3.5) hold for an arbitrary approximation ϕ δ to ϕ p satisfying the conditions ϕ δ ∈ D and ϕ δ − ϕ p 2 ≤ δ. By using the formula (3.3), we obtain the differences

Let us estimate the integral
for p, q = 1, 2, s ∈ [−h, 0], then the equality is valid As a result, we have the asymptotic estimate In a similar way we find that By taking into account the resulting estimates, the property of the regularizing operator R, and inequality (3.5) we complete the proof of the theorem.

Asymptotic of regularized solutions
Let H loc be the space of functions defined on the half-line (−∞, −h] restriction of which to any hold for an arbitrary approximation ϕ δ ∈ D to the initial function. Here N coincides with an integer part of the number |t − |/h, ||ϕ|| = (ϕ, ϕ) 1/2 . For the regularizing operator R the last sum can be made arbitrarily small. Consequently, in the problem of finding solutions of the system (1.1) for any t − < −h the map D → H t − defined by the formula ϕ → x(·, ϕ, δ) is regularizing. The functions x(·, ϕ, δ) ∈ H loc are referred to as regularized solutions of the system (1.1) on the negative half-line.
For the initial function ϕ ∈ W N +1 ∞ [−h, 0], N ≥ 2, we introduce the sequence of functions By using this sequence, we define new sequences . Let us introduce the functions x 1 (·, ϕ), x 2 (·, ϕ, δ) ∈ H t − , by the formulas hold for an arbitrary approximation ϕ δ to ϕ p satisfying the conditions ϕ δ ∈ {ϕ : ∆(ϕ m ) = 0}, For the regularizing operators R 1 and R 2 the last sum can be made arbitrary small. The proof of the theorem is complete.

Example
In the report [2] one can find the following statistic data of the population size of elk in the Vologda region For the analysis of these data we use the obtained results of the paper. Firstly, we solve an identification problem for the Hutchinson-Wright equation. The breeding age of elks h = 2.5 [10, p. 231]. For the finding of the Malthusian factor r and the capacity of the habitat k we use the formula We choose the initial moment t 0 = 2004, and we approximate the initial function ψ on the interval [2001.5, 2004] by cubic splines. Then, by using the above statistical data we obtain By following the posed approach in this work, on the first step of applying the method of reconstruction of the prehistory we find γ(ϕ) = 970.173, ∆(ϕ) = 2459.48. Then the value of the first regularizing operator is defined by the formulas The figure shows the results of computation for the value δ = 10 −2 of the admissible error. The graph of the modeled initial function is given by the black line and the graph of the asymptotic regularized solution x 2 (·, δ, ϕ) by the grey line, and the statistical data is given by black points. The error of reconstruction of the prehistory is defined by the formula j=1999 (x 2 (j, δ, ϕ) − N (j)) 2 = 0.11.