Uncertain population model with age-structure

1 Introduction

 Partial differential equations have wide applications in the life sciences. For example, the motion of organisms, cells, and population dynamics are usually described by some partial differential equations. The extensive applications of partial differential equations to life science problems can be found in [1, 5, 15]. However, these types of partial differential equation models are mostly deterministic in which the uncertain factors are not considered. In this paper, we will present an uncertain population model by an uncertain partial differential equation to describe the population dynamics with the age structure under the framework of uncertainty theory.

 Uncertainty theory was founded by Liu [7] in 2007 for rationally dealing with belief degrees associated with human uncertainty. For representing the uncertain quantities, Liu [7] proposed an concept of uncertain variable, and for modelling the evolution of phenomena with uncertainty, Liu [8] introduced the concept of uncertain process. Especially, an important type of uncertain process which is called canonical Liu process was investigated by Liu [8], and then uncertain calculus with respect to this type of process was established by Liu.

 In 2008, Liu proposed a type of uncertain differential equation driven by canonical Liu process and it has become an important tool to deal with the problems in dynamical systems with uncertainty. For example, uncertain differential equation was first time employed to model stock price by Liu [9], and European option, American option, and Asian option pricing formulas were derived by Liu [9], Chen [2], Sun and Chen [16] and Zhang and Liu [20] based on this type of uncertain stock model, respectively. Subsequently Zhang, Liu and Sheng [23] gave the price formula of power option. Chen and Gao [3] employed uncertain differential equation to model interest rate, following their model, Zhang, Ralescu and Liu [21] investigated the pricing problem of interest rate option. Besides, Liu, Chen and Ralescu [13] derived the currency option pricing formula by using uncertain differential equation to model currency rate. And it also has been applied in many other fields, such as uncertain optimal control (Zhu [24]), uncertain differential game (Yang and Gao [17]).

 Many other types of uncertain differential equations were also proposed by some scholars. For example, Yao [19] introduced uncertain differential equation with jumps, and Li, Peng and Zhang [6] proposed a type of multifactor uncertain differential equation, Zhang, Gao and Yang [22] investigated the stability of this type of multifactor uncertain differential equation. Considering the case of some dynamical systems are usually varying with multiple indexes, Yang and Yao [18] presented a new type of uncertain differential equation: uncertain partial differential equation, and introduced its applications with to heat conduction problems.

 In this paper, we will employ uncertain partial differential equation to model population dynamics, and make some species analysis with it. The rest of this paper is organized as follows. The next section will introduce some preliminaries about uncertainty theory. In Section 3, uncertain partial differential equation model on age structure of population is proposed. In Section 4, some analysis on population dynamics are made. A brief summary is given in Section 5.

2 Preliminaries

 The following is some useful definitions and theorems of uncertainty theory as needed.

Definition 2.1. (Liu [7]) Let Γ be a nonempty set, and let L be a σ-algebra over Γ. An uncertain measure is a function ℳ : L → [ 0 , 1 ] such that

Axiom 1. ℳ {Γ} =1 for the universal set Γ;

Axiom 2. (Duality Axiom) ℳ {Λ} + ℳ {Λ ^c } =1 for any event Λ;

Axiom 3. (Subadditivity Axiom) For every countable sequence of events {Λ _i } we have ℳ { ⋃ i = 1 ∞ Λ i } ≤ ∑ i = 1 ∞ ℳ { Λ i } .(2.1)

 A set Λ ∈ L is called an event. The uncertain measure ℳ {Λ} indicates the degree of belief that Λ will occur. The triplet ( Γ , L , ℳ ) is called an uncertainty space. In order to obtain an uncertain measure of compound event, a product uncertain measure was defined by Liu [9].

Axiom 4. (Product Axiom) Let ( Γ k , L k , ℳ k ) be uncertainty spaces for k = 1, 2, ⋯ The product uncertain measure ℳ is an uncertain measure on the product σ-algebra L 1 × L 2 × ⋯ satisfying ℳ { ∏ k = 1 ∞ Λ k } = ⋀ k = 1 ∞ ℳ k { Λ k }(2.2) where Λ _k are arbitrarily chosen events from L k for k = 1, 2, ⋯, respectively.

Definition 2.2. (Liu [7]) An uncertain variable is a measurable function from an uncertainty space ( Γ , L , ℳ ) to the set of real numbers, i.e., {ξ ∈ B} is an event for any Borel set B.

Definition 2.3. (Liu [7]) The uncertainty distribution Φ of an uncertain variable ξ is defined by Φ ( x ) = ℳ { ξ ≤ x }(2.3) for any real number x.

Definition 2.4. (Liu [7]) An uncertain variable ξ is called normal if it has a normal uncertainty distribution Φ ( x ) = ( 1 + exp ( π ( e - x ) 3 σ ) ) - 1(2.4) denoted by N ( e , σ ) where e and σ are real numbers with σ > 0.

Definition 2.5. (Liu [10]) An uncertainty distribution Φ (x) is said to be regular if it is a continuous and strictly increasing function with respect to x at which 0 < Φ (x) <1, and lim x → - ∞ Φ ( x ) = 0 , lim x → + ∞ Φ ( x ) = 1 .(2.5)

Definition 2.6. (Liu [10]) Let ξ be an uncertain variable with regular uncertainty distribution Φ (x). Then the inverse function Φ^-1 (α) is called the inverse uncertainty distribution of ξ.

Definition 2.7. (Liu [7]) Let ξ be an uncertain variable. Then the expected value of ξ is defined by E [ ξ ] = ∫ 0 + ∞ ℳ { ξ ≥ r } dr - ∫ - ∞ 0 ℳ { ξ ≤ r } dr(2.6) provided that at least one of the two integrals is finite.

Theorem 2.1. (Liu [7]) Let ξ be an uncertain variable with uncertainty distribution Φ. If the expected value exists, then E [ ξ ] = ∫ 0 + ∞ ( 1 - Φ ( x ) ) dx - ∫ - ∞ 0 Φ ( x ) dx .(2.7)

Theorem 2.2. (Liu [10]) Let ξ be an uncertain variable with regular uncertainty distribution Φ. Then E [ ξ ] = ∫ 0 1 Φ - 1 ( α ) d α .(2.8)

Theorem 2.3. (Liu [10]) Let ξ₁, ξ₂, ⋯ , ξ _n be independent uncertain variables with regular uncertainty distributions Φ₁, Φ₂, ⋯ , Φ _n , respectively. If the function f (x₁, x₂, ⋯ , x _n ) is strictly increasing with respect to x₁, x₂, ⋯ , x _m and strictly decreasing with respect to x_m+1, x_m+2, ⋯ , x _n , then the uncertain variable ξ = f ( ξ 1 , ξ 2 , ⋯ , ξ n )(2.9) has an inverse uncertainty distribution Ψ - 1 ( α ) = f ( Φ 1 - 1 ( α ) , ⋯ , Φ m - 1 ( α ) , Φ m + 1 - 1 ( 1 - α ) , ⋯ , Φ n - 1 ( 1 - α ) ) .(2.10)

 Liu and Ha [12] proved that the uncertain variable ξ = f (ξ₁, ξ₂, ⋯ , ξ _n ) has an expected value E [ ξ ] = ∫ 0 1 f ( Φ 1 - 1 ( α ) , ⋯ , Φ m - 1 ( α ) , Φ m + 1 - 1 ( 1 - α ) , ⋯ , Φ n - 1 ( 1 - α ) ) d α .(2.11)

 An uncertain process is a sequence of uncertain variables indexed by a totally ordered set T. A formal definition is given below.

Definition 2.8. (Liu [8]) Let ( Γ , L , ℳ ) be an uncertainty space and let T be a totally ordered set (e.g. time). An uncertain process is a function X _t  (γ) from T × ( Γ , L , ℳ ) to the set of real numbers such that {X _t ∈ B } is an event for any Borel set B of real numbers at each time t.

Definition 2.9. (Liu [11]) Uncertain processes X_1t, X_2t, ⋯ , X _nt are said to be independent if for any positive integer k and any times t₁, t₂, ⋯ , t _k , the uncertain vectors ξ i = ( X it 1 , X it 2 , ⋯ , X it k ) , i = 1 , 2 , ⋯ , n(2.12) are independent, i.e., for any k-dimensional Borel sets B₁, B₂, ⋯ , B _n , we have ℳ { ⋂ i = 1 n ( ξ i ∈ B i ) } = ⋀ i = 1 n ℳ { ξ i ∈ B i } .(2.13)

Definition 2.10. (Liu [9]) An uncertain process C _t is said to be a canonical Liu process if

C₀ = 0 and almost all sample paths are Lipschitz continuous,

 In order to deal with the integration and differentiation of uncertain processes, Liu [9] proposed an uncertain integral with respect to canonical Liu process.

Definition 2.11. (Liu [9]) Let X _t be an uncertain process and C _t be a canonical Liu process. For any partition of closed interval [a, b] with a = t₁ < t₂ < ⋯ < t_k+1 = b, the mesh is defined as Δ = max 1 ≤ i ≤ k | t i + 1 - t i | .(2.14)

 Then the Liu integral of X _t is defined as ∫ a b X t dC t = lim Δ → 0 ∑ i = 1 k X t i ( C t i + 1 - C t i )(2.15) provided that the limit exists almost surely and is finite. In this case, the uncertain process X _t is said to be Liu integrable.

Definition 2.12. (Chen and Ralescu [4]) Let C _t be a canonical Liu process and let Z _t be an uncertain process. If there exist uncertain processes μ _t and σ _t such that Z t = Z 0 + ∫ 0 t μ s ds + ∫ 0 t σ s dC s(2.16) for any t ≥ 0, then Z _t is called a Liu process with drift μ _t and diffusion σ _t . Furthermore, Z _t has an uncertain differential dZ t = μ t dt + σ t dC t .(2.17)

 Liu [9] verified the fundamental theorem of uncertain calculus, i.e., for a canonical Liu process C _t and a continuous differentiable function h (t, c), the uncertain process Z _t  = h (t, C _t ) is differentiable and has a Liu differential dZ t = ∂ h ∂ t ( t , C t ) dt + ∂ h ∂ c ( t , C t ) dC t .(2.18)

Definition 2.13. (Liu [8]) Suppose C _t is a canonical Liu process, and f and g are two functions. Then dX t = f ( t , X t ) dt + g ( t , X t ) dC t(2.19) is called an uncertain differential equation.

 Suppose that the number of some species between age y and y + Δ y at time t is approximately P ( y , t ) Δ y ,(3.1) in other words, P (y, t) is the age density of the population. Suppose that the death number in the time interval from t to t + Δ t is D ( y , t ) P ( y , t ) Δ y Δ t(3.2) where D (y, t) is the per capita mortality rate. Then P ( y , t + Δ t ) = P ( y - Δ t , t ) - ∫ 0 Δ t D ( y - Δ t + s , t + s ) P ( y - Δ t + s , t + s ) ds .(3.3) Subtracting P (y, t) in the two sides of the above equation, then dividing two sides of the equation by Δt and let Δt → 0, we have ∂ P ( y , t ) ∂ t = - ∂ P ( y , t ) ∂ y - D ( y , t ) P ( y , t ) .(3.4)

 For simplicity, we rewrite it as following form P t = - P y - D ( y , t ) P(3.5) or P t + P y + D ( y , t ) P = 0 .(3.6)

 However, the mortality rate can not be deterministic since it is influenced by various uncertain factors, so we add an uncertain noise to the term of D (y, t), that is the term D (y, t) is replaced by D ( y , t ) + σ dC t dt ,(3.7) where D (y, t) can be seen as expected mortality rate, and σ is the volatility with respect to the mortality rate. Then we have the uncertain population model with age-structure as follows P t + P y + ( D ( y , t ) + σ dC t dt ) P = 0 ,(3.8) which is described by the above uncertain partial differential equation.

Theorem 3.1. The uncertain partial differential equation P t + P y + ( D ( y , t ) + σ dC t dt ) P = 0(3.9) has a solution P ( y , t ) = P ( y - t , 0 ) exp [ - ∫ 0 t D ( y - t + s , s ) ds - σ C t ](3.10)in the case of y > t, where P (y - t, 0) is the initial species density of age y - t at time 0.

Proof. Now we write f ( y , t ) = P ( y - t , 0 ) exp [ - ∫ 0 t D ( y - t + s , s ) ds ] .(3.11)

 Then we have f t + f y + D ( y , t ) f = 0(3.12) and P ( y , t ) = f ( y , t ) exp ( - σ C t ) .(3.13)

 Since dP ( y , t ) dt = df ( y , t ) dt exp ( - σ C t ) + f ( y , t ) ( - σ dC t dt ) exp ( - σ C t ) ,(3.14) and dP ( y , t ) dy = df ( y , t ) dy exp ( - σ C t ) .(3.15)

 Combining with (3.12), (3.14) and (3.15), we have dP ( y , t ) dt + dP ( y , t ) dy + ( D ( y , t ) + σ dC t dt ) P ( y , t ) = 0 .(3.16)

 Therefore (3.10) is the solution of Equation (3.9). □

Theorem 3.2. The solution of the uncertain partial differential equation P t + P y + ( D ( y , t ) + σ dC t dt ) P = 0(3.17) has expected value E [ P ( y , t ) ] = P ( y - t , 0 ) exp [ - ∫ 0 t D ( y - t + s , s ) ds ] ∫ 0 1 exp ( - σ t 3 π ln α 1 - α ) d α .(3.18)

Proof. It follows from Theorem 3.1 that E [ P ( y , t ) ] = P ( y - t , 0 ) exp [ - ∫ 0 t D ( y - t + s , s ) ds ] E [ exp ( - σ C t ) ] .(3.19)

 Since C _t has an inverse uncertainty distribution Φ t - 1 ( α ) = t 3 π ln α 1 - α(3.20) and J (x) = exp(- σx) is a decreasing function, we have E [ exp ( - σ C t ) ] = ∫ 0 1 exp ( - σ Φ t - 1 ( 1 - α ) ) d α = ∫ 0 1 exp ( - σ t 3 π ln 1 - α α ) d α = ∫ 0 1 exp ( - σ t 3 π ln α 1 - α ) d α .(3.21)

 Then the theorem is verified. □

Example 3.1. Consider the uncertain partial differential equation P t + P y + ( yt 100 + 0.01 dC t dt ) P = 0(3.22) with the initial condition P ( y - t , 0 ) = 10000 + 10 ( y - t ) .(3.23)

 By Theorem 3.1, we can get P ( y , t ) = [ 10000 + 10 ( y - t ) ] exp ( - yt 2 200 + t 3 600 - 0.01 C t )(3.24) and by the Theorem 3.2, its expected value is E [ P ( y , t ) ] = [ 10000 + 10 ( y - t ) ] exp ( - yt 2 200 + t 3 600 ) ∫ 0 1 exp ( - 0.01 t 3 π ln α 1 - α ) d α .(3.25)

 In many situations, the number of some species in the future is needed to be predicted, it can be regarded as a basis for making a plan or making decisions.

Example 4.1. Suppose the population density at age 60 now is known, we want to know the population density after ten years by which we can make reasonable policies of old-age pension, public service and so on. Suppose the population density at age 60 and time 0 is P (60, 0) =1000000, the mortality rate D ( y , t ) = y 10000 and the volatility is σ = 0.01. Following Theorems 3.1 and 3.2, we can obtain the expected population density at age 70 after ten years is E [ P ( 70 , 10 ) ] = 992800 .

Remark 4.1. Since P (y - t, 0) is not defined for y < t (i.e. negative age), so the solution (3.10) is also not defined for y < t. For y < t, we have following results:

Theorem 4.1. In the case of y < t, the uncertain partial differential equation P t + P y + ( D ( y , t ) + σ dC t dt ) P = 0(4.1) has a solution P ( y , t ) = P ( 0 , t - y ) exp [ - ∫ t - y t ( D ( y - t + s , s ) + σ dC s ds ) ds ](4.2)where P (0, t - y) is the birth rate at time t - y.

Remark 4.2. Note that P (0, t - y) is unknown, here we present a model about birth rate P (0, t) as following dP ( 0 , t ) = ( m - aP ( 0 , t ) ) dt + δ d C ˜ t ,(4.3) where C ˜ t is another canonical Liu process which is independent of C _t . Thus the case of y < t can be dealt with.

 Considering the influence of uncertain factors, for dealing with the problems of population dynamics with uncertainty, an uncertain partial differential equation model of population with age-structure was presented in this paper. Some species analysis and applications based on this model were given.