1. Introduction
The impact of environmental noise on population systems has long been recognized. In previous studies on stochastic population systems [
1,
2], the focus was primarily on modeling changes caused by random factors during death. However, interactions between species or random fluctuations in the environment were not considered. In 2004, Zhang et al. [
3] introduced the Wiener process to the age-dependent model without diffusion and demonstrated the existence, uniqueness, and exponential stability of the stochastic APDS. Subsequently, in 2005, Zhang et al. [
4] conducted numerical analysis on the stochastic APDS. In 2006, Li et al. [
5] performed numerical simulations of the model using Euler’s approximate solution and compared it with the analysis solution without perturbation and the alternative explicit solution. For more models on using the Euler-Maruyama approximation method to deal with stochastic differential systems driven by Wiener process, please refer to Refs. [
6,
7], Moreover, in recent decades, a significant and expanding body of literature has emerged, investigating models and numerical algorithms for stochastic age-dependent population equations, such as [
8,
9].
In [
3,
4,
5], the stochastic age-dependent model without diffusion is as follows:
where
stands for age,
stands for time,
stands for the population density of age
at time
k,
stands for the mortality rate of age
at time
k,
stands for the fertility rate of females of age
at time
k,
stands for influence of external environment for population system.
is the differential of
N related to time k, that is,
,
is Wiener process.
Uncertainty theory [
10] provides a useful account of how to deal with uncertain disturbance related to belief degrees of indeterminate phenomena. In order to describe the evolution of an uncertain phenomenon related to belief degrees, the Liu process [
11] was first applied in uncertain differential equations (UDEs for short) [
12]. After that, Yao et al. [
13] first provided a theorem on Lipschitz continuity of process, based on which, it gave a sufficient condition for an uncertain differential equation being stable. Furthermore, Yao and Chen [
14] first developed the Yao-Chen formula to simulate UDEs. In recent years, they have been successfully applied to many directions such as differential game [
15,
16], partial differential equation [
17] and age-dependent population [
18], fractional differential equation related to symmetry [
19] and so on. The definition of Liu process
is as follows
(a) and almost all sample paths are Lipschitz continuous,
(b) has stationary and independent increments,
(c) every increment
is a normal uncertain variable with expected value 0 and variance
, whose uncertainty distribution is
and inverse distribution with belief degree
is
Randomness and uncertainty are frequently prescribed for indeterminacy and have been given more and more attention. For a complex hybrid system, recently, there has been new interest in chance space [
20], which can effectively deal with the issue of randomness and uncertainty. Uncertain stochastic process is a sequence of uncertain random variables indexed by time. With demand of practical issues need, recently investigators have examined the effects of uncertain random elements on the differential equation system. In 2014, similar to the analysis method of the fuzzy random process, Fei [
21] first considered uncertain stochastic differential equations (USDEs), proved the Itô-Liu formula and applied it to control systems with Markovian switching. In the same year, Fei [
22] proved the existence and uniqueness theorems of backward USDEs. Of course, there are many papers about hybrid systems, such as [
23,
24,
25].
However, none of the previous studies have attempted to investigate the age-dependent population dynamic with uncertainty related to belief degree and randomness at the same time. Consequently, this paper attempts to establish the theoretical research about the hybrid age-dependent population dynamic. Suppose that
is uncertain stochastically perturbed, with
where
is a Wiener process and
is a Liu process, both of which are environmental disturbances. Then, we use the following Itô-Liu equation to describe environmentally perturbed population system.
A new uncertain stochastic model (
2) for APDS is obtained, and we call it hybrid APDS.
The main innovation of this paper is that we first consider a hybrid APDS under chance space. Particularly, we complete the proofs of the existence, uniqueness and stability for a hybrid APDS (
2). The proofs of existence and uniqueness are similar to the Refs. [
3,
26], and the proof of exponential stability is similar to the Refs. [
3,
27].
In
Section 2, we recall some results about Hilbert space, and some concepts, theorems and lemmas about chance space and filtered chance space, which are essential for our analysis. In
Section 3, we first give the integral equation of hybrid age-dependent population Equation (
2), then prove existence, uniqueness and exponential stability of a strong solution for hybrid APDS (
2), respectively.
2. Preliminaries
We will use the variational method, see the Refs. [
3,
5,
26]. Let
where
is generalized partial derivatives. Z is a Sobolev space.
satisfies
where
is the dual space of Z.
,
and
stands for the norms in
Z,
H and
respectively;
stands for the duality product between
Z,
, and
stands for the scalar product in
H. The constant
c satisfies the following inequality:
Let
be the space of all bounded linear operators from
S into
H. For
, we use
to represent the Hilbert-Schmidt norm.
G denotes a symmetric positive definite matrix, that is,
Let
be a nonempty set, and
a
-algebra over
. Each element
in
is called an event and
is the belief degree. Uncertain measure [
10] dealing with belief degree satisfies the following axioms:
Remark 1. Axioms 1 and 2 are similar to probability theory while axioms 3 and 4 are fundamentally different from probability theory. In particular, axiom 3 embodies subadditivity, which is different from the additivity of probability theory, and the product axiom of axiom 4 embodies the minimization operation, which is different from the product axiom of probability theory. The detailed analysis can be found in Ref. [10]. Definition 1 ([
10])
. An uncertain variable is a measurable function ξ from an uncertainty space to the set of real numbers, i.e., for any Borel set B of real numbers, the setis an event. Definition 2 ([
10])
. Let T be an index set and an uncertainty space. An uncertain process is a measurable function from to the set of real numbers such that is an event for any Borel set B for each time k. Let be a complete probability space with a filtration satisfying the usual conditions, that is, it is increasing and right continuous while contains all P-null sets.
Let
be an uncertainty space [
10] where normality, duality, subadditivity and product measure axioms are given. Let
be a Liu process defined on
. The process filtration
is the sub-
-field family (
) of
satisfying the usual conditions. It is generalized by
and
-null sets of
,
.
Liu [
20] first introduced chance theory to investigate hybrid systems with uncertainty with respect to belief degree and randomness. To investigate the uncertain stochastic differential systems, Fei [
22] extended a filtered chance space
on which some concepts, theorems and lemmas are given as follows.
Definition 3 ([
22])
. (a) Let B be a Borel set, an uncertain random variable ξ is a measurable function from a chance spaceto (or , or H), that is, (or ,or H), the set(b) The definition of the expected value iswhere and denote the expected values under the probability space and the uncertainty space, respectively. (c) , is an uncertain random event with chance measure Clearly, for constant and , we have where is a one-dimensional Wiener process and is a scalar process.
Proposition 1 ([
28])
. (Linearity of Expected Value Operator) Assume and are (not necessarily independent) random variables, and are independent uncertain variables, and and are measurable functions. Then, Definition 4. Assume η is a random variable, τ is an uncertain variable with the regular uncertainty distribution Ψ and a measurable function f, suppose is strictly increasing as , is strictly decreasing as , and denote as the inverse function of as and as the inverse function of as . The generalized expected value of is defined as Remark 2. If satisfies , then we can easily obtain by the subadditivity of the uncertain random measure.
Definition 5 ([
22])
. (a) For each time if is an uncertain random variable, then we call an uncertain stochastic process. If the sample paths of are continuous functions of k for almost all then we call it continuous.(b) If is -measurable for all then we call it -adapted. Further, if is -measurable for all then we call it -adapted (or adapted).
(c) If the uncertain stochastic process is measurable related to the σ-algebrathen we call it progressively measurable. Further, if the uncertain stochastic process (or is progressively measurable and satisfies , then we call it -progressively measurable, where (or )) denotes the set of -progressively measurable uncertain random processes. The extension of or to a Hilbert space can be omitted here. Later, we will give the definition in the next section.
Next, we redefine the Liu integral in the mean square sense according to Remark 2 of Ref. [
29].
Definition 6 ([
29])
. Let be an uncertain process and be a Liu process. For any partition of closed interval with , the mesh is written asThen uncertain integral of with respect to isprovided that the limit exists in the mean square sense and is finite. In this case, the uncertain process is said to be integrable. Definition 7 ([
21])
. Let be a Wiener process and a Liu process. Then, is called a Wiener-Liu process. The Wiener-Liu process is said to be standard if both and are standard. Definition 8 ([
21])
. Let where and are scalar uncertain stochastic processes, and let be a standard Wiener-Liu process. For any partition of closed interval with , the mesh is written asThen uncertain stochastic integral of with respect to isprovided that the limit exists in mean square and is a uncertain random variable. Remark 3. The uncertain stochastic integral may also be written as follows: The following theorem will give the Itô-Liu formula of the 1-dimensional case.
Proposition 2 ([
21])
. Let be a Wiener-Liu process given byLet be a Wiener process and a Liu process and a twice continuously differentiable function. Define Then, we have the following chain rule: Corollary 1 ([
21])
. The infinitesimal increments and in (28) may be replaced with the derived Wiener-Liu process,where and are absolutely integrable uncertain stochastic processes, and is a square integrable uncertain stochastic process, then , ( means second order continuous differentiable), thus producing Let
be a p-dimensional standard Wiener process, and
a q-dimensional standard Liu process. If
and
are absolute integrable hybrid processes, and
are square integrable hybrid processes, for
then the m-dimensional hybrid process
is given by
Or, in matrix notation, simply,
where
Proposition 3 ([
21])
. Assume m-dimensional hybrid process is given byLet be a multivariate continuously differentiable function. Define . Then, where , , for Here, In other words, it can be expressed as
By Pardoux [
30] and Ichikawa [
31], the extension of the general Itô-Liu formula can be omitted here.
Lemma 1 ([
12] (Liu’s lemma))
. Let be a Liu process, and be an integrable uncertain process on with respect to time k. Then, the inequalityholds, where is the nonnegative Lipschitz constant of the sample path . Corollary 2 (the extension of Liu’s lemma)
. Let be a Liu process, and be an integrable uncertain stochastic process on with respect to k, and Then, the inequalityholds, where L is a nonnegative constant. Proof. Similar to the proof of Lemma 1, we can obtain
where
is the nonnegative Lipschitz constant of the sample path
, then
is an uncertain variable on
; thus,
So, taking the expected values on both sides, we obtain
In addition, by Holder’s inequality [
10],
By using Theorem 2 of Ref. [
13], we know that
Since
is convergent, so
is convergent. Consequently, it holds that
where, we simply denote
by
L. So, it holds that
The proof of the corollary is completed. □
Lemma 2 ([
21])
. By using the Itô-Liu uncertain stochastic integral and the expectations operator , then, , , and ,where space can extend to Hilbert space, see [32] (omitted here). We will use these results in the next section. Lemma 3 (Gronwall inequality [
33])
. Let be a nonnegative function such asfor some constants . Then, we have 3. Main Results
In this paper, let be the space of all continuous functions from into H with sup-norm , and .
Here, we give the equivalent uncertain stochastic integral Equation (
5) of (
2):
where
,
.
Definition 9. Suppose that is an uncertain random variable such that . An uncertain stochastic process is said to be a strong solution on to the uncertain stochastic integral Equation (5) for if the following hold: (a) is a -measurable uncertain random variable in time k (we denote by in the sequel);
(b) , , where stands for the space of all Z-valued processes ( for short) measurable (from into Z), and satisfying Here, stands for the space of all continuous functions from to H;
(c) , Equation (5) is satisfied with chance measure one. If T is replaced by ∞, is called a global strong solution of (5). Let
Q denote
. We will next prove that there exists a unique
such that (
5) holds. Firstly, we give some assumptions:
Let be a family of nonlinear operators defined a.e.k., and they satisfy (c.1) and Lipschitz condition (c.2):
(c.1) ;
(c.2)
such that
Let be a family of nonlinear operators defined a.e.k., , and they satisfy (c.3) and Lipschitz condition (c.4)
(c.3)
(c.4) such that
,
(c.5)
,
are nonnegative measurable, and
(c.6) There exist constants
,
,
, and a nonnegative continuous function
,
, such that
Here, , satisfies , as i.e., .
3.1. Uniqueness Theorem
Theorem 1. (Uniqueness) Assume (c.1)–(c.5) hold. Then, there exists a unique solution of (5) in Q. Proof. Suppose that
are two solutions of (
5). Then, for
, it holds from Itô-Liu’s formula that
because
Then,
, by (c.2) and (c.4), it holds that
By BDG inequality, it holds that
By the extension of Liu’s lemma (Corollary 2), it holds that
where
are positive constants.
Substituting (
8) and (
9) into (
7) yields
By Gronwall inequality (Lemma 3), we get the uniqueness. □
3.2. Existence Theorem
We firstly consider the following equations
Remark 4. These equations are a sequence of successive approximations of functions. We will prove the existence theorem by a successive approximation method.
Before we prove the existence theorem, we first prove three lemmas.
Lemma 4. Assume the conditions hold. Then, is a Cauchy sequence in .
Proof. For
and the uncertain stochastic process
, by Itô-Liu’s formula, it holds that
where, by definition,
,
,
and
. It is easy to deduce
Consequently, (
14) yields
Next, we will estimate the five terms of (
15) by using the inequality
By the BDG inequality, it holds that
By the extension of Liu’s lemma (Corollary 2), it holds that
where
K is a constant, and we denote
combining with (
15)–(
19), it holds that
where
is a positive constant, so there exists
such that
By iteration from (
22), we obtain
By (
24), it is clear that
is a Cauchy sequence
. □
Lemma 5. Assume the conditions hold. Then, the sequence is bounded in .
Proof. By applying Itô-Liu’s formula to
with
, we can obtain
By (c.6), the following inequalities hold:
We can see that, owing to the subexponential growth and continuity of the
,
, there exists a positive constant
such that
. Combining with (
26)–(
29), we can deduce the following inequality.
Because
is convergent in
, which leads to the boundedness of
. Thus, there exists a constant
such that
and Lemma 5 is proved. □
Lemma 6. Assume the conditions (c.1)–(c.6), and hold. Then, there exists a unique process such thatwhere , and is an H-valued continuous, square integrable -martingale. Proof. See the Refs. [
26,
32] and, similar to Proposition 3.1 in [
22], it is easy to get the Lemma 6. □
Next, we give the proof of the existence of a solution to the problem (
5).
Theorem 2. (Existence) Assume (c.1)–(c.4) and (c.6) hold. Then, for each , there exists a unique solution of Equation (5) in Proof. The family
, defined as
, satisfies the assumptions in Lemma 6. Combining with (c.1), as a result, (
10)–(
12) have a unique solution
.
For
, it follows from (c.2), (c.4) and Lemma 6 that there exists a unique process
of (
10)–(
12).
By recurrence, it holds that
is a unique solution of (
10)–(
12).
Next, we will prove that the sequence
is convergent to a process
in
Q, which will be the solution of (
5).
Firstly, by Lemma 4, it holds that there exists
such that
in
. By (c.2) and (c.4), we can obtain
, and
in
,
in
.
Through the previous analysis, it is easily to get that
In addition, according to Lemma 5, has a subsequence which is weakly convergent in . Owing to in , it holds that weakly in (denoted by ), and
Owing to the continuity of the differential operator D, . Combining with the proof of uniqueness of Theorem 1, we complete the proof of Theorem 2. □
3.3. Stability Theorem
Theorem 3. Assume and hold. If is a global strong solution to Equation (5), then there exist constants , such that Proof. Take small enough
such that
. By Itô-Liu’s formula, it holds that
By Lemma 2, it follows that
and
By condition (c.6) and (
3) yield
where
.
If
, it follows immediately
Then, there exists a positive constant
such that
If
, we take small enough
such that
. Thus, by (
33) and Gronwall’s lemma (Lemma 3), we can obtain
So, there exists a positive constant
such that
Next, we will give a generalization of Theorem 3.
(c.7) There exist constants
,
,
,
and a nonnegative continuous function
,
, such that
Here, , satisfy , . □
Theorem 4. Assume and hold. If is a global strong solution to Equation (5), there exist constants , such that Proof. Similar to the proof of Theorem 3, take small enough
such that
. By Itô-Liu’s formula, it holds that
By
, (
3), it holds that
where
.
If
, it follows immediately
By the extended Gronwall-type Lemma [
27], it holds that
Then, there exists a positive constant
such that
If
, we take small enough
such that
. So, by (
39), we can obtain
By the Gronwall’s lemma (Lemma 3), we can obtain
By the extended Gronwall-type Lemma again, it holds that
□
Remark 5. For the finite dimensional case, such as one dimensional case, we take , . As a result, is a one-dimensional Wiener process, and is a one-dimensional Liu process.
Remark 6. Different from existence, uniqueness and exponential stability of stochastic age-dependent population equation based on probability theory of additive measure [3,4], uncertain stochastic hybrid age-dependent population equations are more complex in terms of dealing with conditions and calculation of existence, uniqueness and exponential stability, such as the conditions of Lemmas 4–6, Theorems 1–4. In addition, we use the Itô-Liu formula, the Liu lemma (Lemma 1), the extended Liu’s lemma (Corollary 2), etc. And these conclusions are all obtained under subadditive measures. Specifically, Refs. [3,4] can be seen as the special case of the model proposed in this paper. 4. Conclusions
The main objective of this study is to propose a new and innovative type of hybrid APDS. In doing so, several significant contributions have been made. Firstly, we redefine the Liu integral in a more comprehensive manner, specifically in terms of its mean square sense. This refined definition enhances our understanding of the integral and its application in the context of the hybrid APDS. Moreover, this study extends the Liu’s lemma and the Itô-Liu formula, expanding their scope and applicability to incorporate the hybrid APDS. By extending these fundamental mathematical tools, we are able to provide a more comprehensive framework for analyzing and understanding the dynamics of the hybrid APDS. Furthermore, an essential aspect of this study is confirming the existence, uniqueness, and stability of the hybrid APDS based on subadditive measures. Through rigorous mathematical analysis and proof, we establish the fundamental properties and characteristics of the hybrid APDS based on subadditive measure, providing a solid theoretical foundation for future research. Overall, this study significantly advances our understanding of the hybrid APDS by redefining key mathematical concepts, extending established formulas, and establishing its essential properties. These contributions lay the groundwork for future investigations and enable us to delve deeper into the dynamics and behavior of the hybrid APDS based on subadditive measure in various models.