
Citation: Ruofeng Rao, Xiaodi Li. Input-to-state stability in the meaning of switching for delayed feedback switched stochastic financial system[J]. AIMS Mathematics, 2021, 6(1): 1040-1064. doi: 10.3934/math.2021062
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Single species population dynamics are governed by the growth rate, which is further determined by the survival and reproduction of its individuals. Both the survival and reproduction rates differ from individual to individual, depending on age, body mass, etc. Trivially speaking, individuals in the reproductive stage directly contribute to the birth rate, and the survival rates vary by life stages. In combination, variations in demographic rates among individuals should be appropriately incorporated into a population growth model. Instead of considering excessively detailed demographic characteristics in laboratory or field experiments, it would be more practical in some cases to lump individuals with similar characteristics together in a specific life stage. On the other hand, ignoring the variations among different stages can lead to misleading predictions on population dynamics [1]; therefore, stage-structured models are proposed as an ideal tool to describe population growth, which take a balance between the model complexity and model performance. Usually, individuals in the same stage can be assumed to undergo an identical development time (a mean development delay) while omitting variances in stage durations. However in some scenarios, the time an individual takes in a specific life stage is not uniformly distributed. For example, some eggs hatch (e.g., become larvae in some insects) before other eggs laid at the same time [1], and a non-uniform (non-Dirac) distribution for the stage duration should be considered when grouping age-stratified individuals together in a stage. To describe the heterogeneity in development, various distributions for stage durations have been fitted from the stage-frequency data obtained from monitoring cohorts through time, including gamma, Weibull, log-normal, logistic and other distributions [2]. For example, the widely used gamma distribution with a positive integer shape parameter n and rate parameter nλ>0 has the following probability density function
f(t;n,nλ)=tn−1e−nλt(nλ)nΓ(n)=tn−1e−nλt(nλ)n(n−1)!. |
This special gamma distribution is also called an Erlang distribution as the shape parameter n is a positive integer. The mean value and variance of this distribution are 1λ and 1nλ2. It has been fitted to the stage-frequency data [2,3] for different species. When n=1, it becomes an exponential distribution. Taking the limit case when n goes to infinity, it becomes the Dirac distribution (also called Dirac δ-distribution). Different probability density functions are illustrated in Figure 1(a) for a mean duration fixed at 1λ=10 days. In this case, the probability of an individual with the stage age a remaining in the particular stage is presented in Figure 1(b).
Competition occurs when two or more individuals of the same or different populations negatively affect each other, striving for limited resources such as food, water, territory, sunlight and mates [4]. There are two different types of competition: intraspecific competition which occurs between individuals of the same species and interspecific competition which occurs between individuals of different species. The logistic growth model incorporates the density-dependent population growth rate due to intraspecific competition, and describes the sigmoid growth curve for a single species. Many generalized forms of the logistic equation have been proposed to fit the observed growth phenomena; examples include, the Richards model for species growth [5] and epidemic data [6], and a more generalized logistic growth model [7,8]. When individual movement in a spatial habitat is considered, the Fisher-KPP equation [9] for a continuous spatial domain and multi-patch logistic questions can be formulated [10]. The spatial dynamics can be investigated and the maximal total population problem can be further studied [11,12,13]. After almost a century of research, the competitive interaction continues to fascinate researchers who want to understand its role in shaping the population dynamics of a single species and engaging species in a community.
The main focus here is on stage-structured models of individuals of the same species competing for the same resources in an ecosystem (e.g. food or living space). However, it should be noted that the stage-structured modeling idea for single population growth has been widely employed in other research areas, such as for disease transmission with various infectious period distributions [14,15,16,17,18] and stage-dependent exposure [19], spatial population dynamics in continuous [20] and discrete [21] habitats, within-host virus dynamics to account for the stages of the viral life cycle before viral production [22], the immune responses of T cell life stages [23] and the waning of the immunity of a vaccinated individual [24].
In this manuscript, two basic modeling approaches based on integral equations and partial differential equations, are reviewed in Section 2. Both frameworks are further reduced into ordinary differential equations with or without a time delay under further Dirac and gamma distribution assumptions on the development time, respectively. Further remarks on advantages and inherent limitations are briefly discussed in the same section. Section 3 is devoted to presenting recent modeling studies, in particular those by Stephen A. Gourley and collaborators, when the mean stage duration and survival probability are regulated by population density due to intraspecific competition. Section 4 concludes the manuscript by discussing some related problems on the topic.
In this section, we will review two basic structured modeling frameworks [25] based on integral equations and partial differential equations. The main focus will be on reducing the models under gamma and Dirac distribution assumptions for stage progression. Without loss of generality, we consider the simplest case when there are two stages, denoted as immature (pre-reproductive) and mature (reproductive) stages with population sizes I(t) and M(t), respectively. The sojourn functions PI(a) and PM(a) describe the probabilities that a living individual remains in immature and mature stages for a units of time (stage age a), respectively, and satisfy the following properties: (ⅰ) 0≤PX(a)≤1; (ⅱ) PX(a) is nonincreasing on a; (ⅲ) ∫∞0PX(a)da<∞, where X=I, M. Please note that we assume the function PX(a) is differentiable, with the derivative P′X(a) applied for the ease of notational simplicity. When it is not differentiable, the Riemann integrals should be represented as Riemann-Stieltjes integrals; rigorous treatments can be found in [26]. Further interesting biological indices can be derived from this sojourn function [26], such as (ⅰ) the mean sojourn time in stage X (mean duration of the stage) can be directly computed as D=∫∞0PX(a)da; (ⅱ) the expected remaining sojourn times at the stage age s would be D(s)=1PX(s)∫∞sPX(η)dη and D=D(0); (ⅲ) the average expectation of the remaining sojourn (duration) E=∫∞0aPX(a)da∫∞0PX(a)da; (ⅳ) the variance of the stage duration is V=D(2E−D).
Population dynamics are intuitively dependent on the stage duration distributions PX(a); the Dirac and gamma distributions will be further discussed in more details. By default, the term "age" represents the chronological age of an individual. In this section, the stage-specific age will be used in some arguments, instead of the chronological age, to measure the time since entering the stage (also called the age within stage). Taking a mature individual with the chronological age η who matures at the chronological age s as an example, this individual develops to the mature stage at I-stage age s and has M-stage age η−s.
Individuals in the immature stage I at a time t include those born at a previous time s, surviving to the time t with the survival probability ΠI(t−s) and staying in the stage with the probability PI(t−s), as well as those introduced at the initial time but remaining alive and in the stage. These arguments lead to the following integral form for the population size of immatures:
I(t)=∫t0B(M(s))⏟bornattimesstayinthestage⏞PI(t−s)ΠI(t−s)⏟survivalds+I0(t)⏟remainingimmatures. | (2.1) |
Here the birth rate B(M(t)) at a time t is a function of the population size of the reproductive stage M(t). The number of immatures that were introduced at the initial time and stay in the stage is
I0(t)=I(0)PI(t)ΠI(t). |
The dynamic evolution of the matured population size M(t) can be depicted in Figure 2 in consideration of birth, survival and stage progression from the previous I stage, as well as the development to a consequential old stage. Individuals in the mature stage M at time t include (ⅰ) those entering the stage at time η, surviving in the stage with the survival probability ΠM(t−η) and staying in the stage with stage distribution function PM(t−η), and (ⅱ) those mature individuals staying in the stage from the initial time or those developed from initially introduced immatures (M0(t) in Eq (2.2)). Please note that individuals entering the M-stage at time η have M-stage age t−η. Furthermore, individuals entering to the mature stage at time η are exactly those born at time s≤η, surviving through the immature stage with the probability ΠI(η−s) and maturing at the time η at rate −P′I(η−s) (these individuals entering the mature stage have the I-stage age η−s). Here, we should mention that the development rate for immature individuals with I-stage age a is given by the derivative −P′I(a). This term in the form of the probability density function can be derived from the following observations: An individual leaves I-stage and enters to the M-stage during the age interval (a,a+Δa) with the probability PI(a)−PI(a+Δa). Taking the limit when Δa goes to zero, the individuals with I-stage age a develop to M-stage at the rate −P′I(a). These arguments give rise to the following equation
M(t)=∫t0∫η0B(M(s))⏟bornatssurvivetoηinI−stage⏞ΠI(η−s)(−P′I(η−s))⏟enterM−stageatηdsPM(t−η)⏟stayinM−stagesurviveinM−stage⏞ΠM(t−η)dη+M0(t)⏟remainedordevelopedfrominitiallyintroducedI | (2.2) |
The term M0(t) can be expressed as
M0(t)=M(0)PM(t)ΠM(t)+I(0)∫t0ΠI(η)(−P′I(η))ΠM(t−η)PM(t−η)dη, | (2.3) |
with M(0)PM(t)ΠM(t) capturing the number of remaining mature individuals introduced at time 0 and I(0)∫t0ΠI(u)(−P′I(u))ΠM(t−u)PM(t−u)du measuring the number of mature individuals developed from immatures introduced at time 0. On the other hand, if we introduce the following term to represent the maturation rate at time η
F(η)=∫η0B(M(s))(−P′I(η−s))ΠI(η−s)ds+I(0)ΠI(η)(−P′I(η)), | (2.4) |
then we can rewrite the Eqs (2.2) and (2.3) to obtain
M(t)=∫t0F(η)⏟maturationrateattimeηsurviveinM−stage⏞ΠM(t−η)PM(t−η)⏟stayinM−stagedη+M(0)PM(t)ΠM(t)⏟remainingmatureindividuals. | (2.5) |
If one regards the maturation rate as the "birth rate" to the mature stage, then M(t) in Eq (2.5) takes a similar form as I(t) in Eq (2.1). We would like to remark that the stage-structured models in integral form go back to Lotka [27]. The model derivation was rigorously presented in [26,Chapter 13] upon careful consideration of the movement through a stage and the stage contents, stage input and stage outputs, which also relaxes the differentiability assumption on PI(a) by using Stieltjes integrals.
Since the stage duration distribution is the main focus of the current study, we take simple exponential distributions for the survival functions ΠI(a) and ΠM(a), that is
ΠI(a)=e−μIaandΠM(a)=e−μMa | (2.6) |
for individuals staying with the stage age a in the I-stage and M-stage, respectively. The parameters μI and μM denote the death rates in immature and mature stages. When the stage duration distributions take some ecologically justified functions, the integral stage-structured model (2.1) and (2.2) can be rewritten into ordinary differential equations with/without time delays. In the following subsections, gamma and Dirac distributions will be discussed. However, we should mention that other stage length distributions would be more appropriate in some scenarios: for example, the distribution of the time duration spanning infection to disease death is better fitted by a lognormal distribution than by a gamma distribution [28] or Weibull distributions. More interesting investigations into log-normally distributed stage durations can be found in [28] and [26,Section 12.8].
Assuming the stage duration follows a gamma distribution, then the probabilities of an individual with the stage age a remaining in each stage are given by
PI(a)=Gnnλ(a)=n∑j=1(nλa)j−1e−nλa(j−1)! | (2.7) |
and
PM(a)=Gmmγ(a)=m∑i=1(mγa)i−1e−mγa(i−1)! | (2.8) |
with shape and rate parameter sets (n,nλ) and (m,mγ) respectively. It should be highlighted that the age a is not the chronological age, but the stage-specific age for the actual amount of time an individual has been alive in the I- and M-stages respectively.
With the exponential survival probability for immatures, we have
I0(t)=I(0)PI(t)ΠI(t)=I(0)e−μItn∑j=1(nλt)j−1e−nλt(j−1)!. |
Substituting (2.7) into the equation of I(t) (Eq (2.1)) gives
I(t)=∫t0B(M(s))e−μI(t−s)n∑j=1(nλ(t−s))j−1e−nλ(t−s)(j−1)!ds+I(0)e−μItn∑j=1(nλt)j−1e−nλt(j−1)!=n∑j=1(∫t0B(M(s))e−μI(t−s)(nλ(t−s))j−1e−nλ(t−s)(j−1)!ds+I(0)e−μIt(nλt)j−1e−nλt(j−1)!)=n∑j=1Ij(t), |
where
Ij(t)=∫t0B(M(s))e−(μI+nλ)(t−s)(nλ(t−s))j−1(j−1)!ds+Ij0(t),1≤j≤n | (2.9) |
with Ij0(t)=I(0)e−(μI+nλ)t(nλt)j−1(j−1)!. Using these sub-stage variables Ij(t), a differential equation system can be derived as follows by taking the derivative of each Ij(t) with respect to t:
I′1(t)=B(M(t))−(nλ+μI)(∫t0B(M(s))e−(μI+nλ)(t−s)ds+I(0)e−(μI+nλ)t)=B(M(t))−(nλ+μI)I1(t), |
and for 2≤j≤n,
I′j(t)=B(M(t))e−μI⋅0(nλ⋅0)j−1e−nλ⋅0(j−1)!+∫t0B(M(s))ddt(e−(μI+nλ)(t−s)(nλ(t−s))j−1(j−1)!)ds+I(0)ddt(e−(μI+nλ)(t−s)(nλt)j−1(j−1)!)=−(μI+nλ)[∫t0B(M(s))(nλ(t−s))j−1(j−1)!ds+I(0)(e−(μI+nλ)t(nλt)j−1(j−1)!)]+nλ[∫t0B(M(s))e−(μI+nλ)(t−s)(nλ(t−s))j−2(j−2)!ds+I(0)e−(μI+nλ)t(nλ(t−s))j−2(j−2)!]=nλIj−1(t)−(nλ+μI)Ij(t). |
Note that the probability density function for the gamma distribution PI(t) satisfies
−P′I(a)=nλ(nλa)n−1e−nλa(n−1)! |
The maturation rate in (2.4) becomes
F(t)=∫t0nλ[(nλa)n−1e−nλa(n−1)!]B(M(t−a))e−μIada−nλ(nλt)n−1e−nλt(n−1)!e−μItI(0)=nλIn(t). |
The equation for the mature stage (2.5) can be rewritten as
M(t)=∫t0F(η)PM(t−η)ΠM(t−η)dη+M(0)PM(t)ΠM(t)=∫t0F(η)e−(μM+mγ)(t−η)m∑i=1(mγ(t−η))i−1(i−1)!dη+M(0)e−(μM+mγ)tm∑i=1(mγt)i−1(i−1)!=m∑i=1Mi(t), |
with
Mi(t)=∫t0F(η)e−(μM+mγ)(t−η)(mγ(t−η))i−1(i−1)!dη+M(0)e−(μM+mγ)t(mγt)i−1(i−1)! |
Similar arguments as those for Ij(t) lead to a series of equations for variables of mature individuals in each sub-stage:
M′1(t)=F(t)−(μM+mγ)M1(t)=nλIn(t)−(μM+mγ)M1(t),M′i(t)=mγMi−1(t)−(mγ+μM)Mi(t), 1<i≤m. |
Therefore, by introducing sub-stage variables Ij(t) and Mi(t), a closed ordinary differential equation model can be derived from the integral form given by (2.1) and (2.2) when the stage duration distributions follow the gamma distributions in (2.7) and (2.8):
dI1(t)dt=B(M(t))−(nλ+μI)I1(t),dIj(t)dt=nλIj−1(t)−(nλ+μI)Ij(t),1<j≤n,dM1(t)dt=nλIn(t)−(μM+mγ)M1(t),dMi(t)dt=mγMi−1(t)−mγMi(t)−μMMi(t),1<i≤m. | (2.10) |
The Dirac stage distribution is suitable to describe the case when individuals entering a specific stage together are assumed to undergo an identical development time that is equal to the mean development delay while the variances in the stage duration are omitted [2]. For the convenience of illustration, we simply assume that PM(ξ)≡1 for all M-stage ages ξ, that is, the mature individuals, if alive, will always stay in the stage. The probability function for immature stage duration with a mean value τ can be expressed as
PI(a)={1,0≤a≤τ0,a>τ. |
This means that alive individuals with I-stage age smaller than τ remain in the I-stage, while those with an age larger than τ develop to M-stage. Please note that PI(a) is not differentiable. However, for notational simplicity, we use the concept of Dirac delta function δ(x) to represent its probability density function. Arguments to deal with non-differentiable sojourn functions can be found in [26,Chapter 13]. We consider the case when t>τ (by resetting the initial timing), and therefore, all immature individuals introduced at time 0 will either die or develop to the M-stage after time t, that is I0(t)=0 for all t>τ. The assumption t>τ also implies M0(t)=e−μMtM(0)+I(0)e−(μIτ+μM(t−τ)). Then we have,
I(t)=∫t0B(M(s))e−μI(t−s)PI(t−s)ds+I0(t)=∫tt−τB(M(s))e−μI(t−s)ds, |
which can be written into a differential equation
I′(t)=B(M(t))−μI∫tt−τB(M(s))e−μI(t−s)ds−B(M(t−τ))e−μIτ=B(M(t))−μII(t)−B(M(t−τ))e−μIτ. |
Then, Eq (2.2) for M(t) now becomes
M(t)=∫t0∫η0B(M(s))(−P′I(η−s))e−μI(η−s)dse−μM(t−η)dη+e−μMtM(0)+I(0)e−(μIτ+μM(t−τ))=∫t0∫η0B(M(s))δ(η−s−τ)e−μI(η−s)dse−μM(t−η)dη+e−μMtM(0)+I(0)e−(μIτ+μM(t−τ))=∫tτB(M(η−τ))e−μIτe−μM(t−η)dη+e−μMtM(0)+I(0)e−(μIτ+μM(t−τ)). |
Please note that δ(⋅) is the corresponding Dirac delta function. Taking the derivative of M(t), we obtain
M′(t)=B(M(t−τ))e−μIτ−μMM(t). |
The integral form can be written as other equivalent forms according to different biological arguments on stage progression, birth and survival. If we consider the stage progression of individuals as illustrated in Figure 3, then the integral form of (2.2) can be written as
M(t)=∫t0B(M(t−η))⏟birthprobabilityofenteringandstayingaliveinM−stagewithchronologicalageη⏞∫η0ΠI(s)⏟survivethroughI−stage(−P′I(s))⏟enterM−stagePM(η−s)ΠM(η−s)⏟aliveandstayinM−stagedsdη+M0(t) |
with M0(t) given in (2.3). In this expression, the integral term accounts for individuals with the chronological age η at time t (i.e., those born at a previous time t−η with η∈[0,t]) and successfully entering and staying in the M-stage alive.
It is easy to obtain the net reproduction number R0 with the following renewal argument for the Volterra integral form (see [29]). In fact, if we assume the population size is very small and the density-dependent regulations on the birth rate function B(M(t)) can be ignored, then the birth rate at time t can be approximated as
B(M(t))≈b⋅M(t) |
with a constant per-capita birth rate b. Let
Φ(η)=b×∫η0−P′I(s)ΠI(s)PM(η−s)ΠM(η−s)ds⏟the probability of an individual with the chronological age η developing to and staying in the M-stage, |
then we have the following Volterra integral form for the population size of the M-stage
M(t)=∫t0M(t−η)Φ(η)dη+M0(t). |
The net reproduction number in demography can be defined as
R0=∫∞0Φ(η)dη. |
Suppose PI(x) takes the gamma distribution function given by (2.7) and PM(x)=1. By taking the exponential survival probability (2.6), we have
Φ(η)=b∫η0−n∑j=1(nλ(nλs)j−2e−nλs(j−2)!−nλ(nλs)j−1e−nλs(j−1)!)e−μIse−μM(η−s)ds=b∫η0nλ(nλ)n−1sn−1(n−1)!e−(nλ+μI−μM)se−μMηds=b⋅e−μMη⋅(nλ)nn!∫η0e−(nλ+μI−μM)sdsn=bn(nλ)ne−μMηn![e−(nλ+μI−μM)ηn−1∑i=0(−1)n−1−i(n−1)!⋅ηii!⋅(−(nλ+μI−μM))n−i−(−1)n−1⋅(n−1)!(−(nλ+μI−μM))n], |
which implies that
R0=bn(nλ)nn!∫∞0e−(nλ+μI)ηn−1∑i=0(−1)n−1−i(n−1)!⋅ηii!⋅(−(nλ+μI−μM))n−idη+b(−nλ)n(−(nλ+μI−μM))n∫∞0e−μMηdη=b⋅(nλ)n⋅n−1∑i=0−1(nλ+μI)i+1(nλ+μI−μM)n−i+bμM⋅(nλ)n(nλ+μI−μM)n=b⋅(nλ)n(nλ+μI)n+1[n−1∑i=0−1(nλ+μI−μMnλ+μI)n−i+(nλ+μI)n+1μM(nλ+μI−μM)n]=bμM(nλnλ+μI)n. |
For the simple case that the immature stage duration follows an exponential distribution with the mean duration 1/λ, namely PI(a)=e−λa, then
R0=bλμM(λ+μI). |
When PI(a) takes the Dirac distribution with the mean duration τ and PM(a)≡1 as those in Section 2.1.2, we have
Φ(η)=b×∫η0−P′I(s)e−μIse−μM(η−s)ds=b×∫η0δ(s−τ)e−μIse−μM(η−s)ds, |
and therefore,
R0=∫∞0Φ(η)dη=∫∞τbe−μIτe−μM(η−τ)dη=be−μIτμM. |
By seeking the solution to the following characteristic equation (Euler-Lotka equation)
∫∞0e−rηΦ(η)dη=1, |
one can determine the initial growth rate (also called intrinsic growth rate or Malthusian parameter [30]). Let f(r)=∫∞0e−rηΦ(η)dη, and assume there exists a real number ˆr such that 1≤f(ˆr)<∞ (for most biological models, we may always find such a negative ˆr). Then it is interesting to observe the following facts: (ⅰ) f(r) is a nonincreasing and continuous function of r, (ⅱ) f(ˆr)≥1 and (ⅲ) limr→∞f(r)=0. Therefore, the above equation f(r)=1 admits a unique real root r=r0∈[ˆr,∞), which is the intrinsic growth rate. Using the identity that R0=f(0) and the monotonicity of f(r), it is evident that the sign of r0 is the same as that of R0−1. Furthermore, the monotonicity f(r) and uniqueness of the real root to the equation f(r)=1 facilitate the design of efficient numerical algorithms, such as the bisection method algorithm, to compute the initial growth rate.
Assume u(a,t) and v(ξ,t) are the population densities of immature and mature individuals at time t with stage-specific ages a and ξ, respectively. Then the sizes of the populations in the immature and mature stages can be expressed as
I(t)=∫∞0PI(a)u(a,t)da and M(t)=∫∞0PM(ξ)v(ξ,t)dξ, |
where PI(a) and PM(ξ) represent the probability functions of individuals with the stage-age a staying in the immature and mature stages, respectively. On the other hand, the following partial differential equation, originally proposed by McKendrick [31] and widely used in recent studies such as [25,32,33,34], can be employed to describe the dynamics of age-dependent variables
∂u(a,t)∂a+∂u(a,t)∂t=−μIu(a,t),u(a,0)=u0(a), | (2.11) |
and
∂v(ξ,t)∂ξ+∂v(ξ,t)∂t=−μMv(ξ,t),v(ξ,0)=v0(ξ) | (2.12) |
with natural death rates μI and μM in each stage.
It is reasonable to assume that the density of the immatures u(0,t) with age 0 at time t is exactly the birth rate, that is
u(0,t)=B(M(t)). |
The density of mature individuals with M-stage age 0 at time t is that of immature individuals developing to the M-stage at time t, that is
v(0,t)=∫∞0[−P′I(a)]u(a,t)da, |
where −P′I(a) represents the development rate for immature individuals with I-stage age a, as discussed in Section 2.1.
When the stage duration follows gamma distributions as given by (2.7) and (2.8), we have
v(0,t)=∫∞0[−P′I(a)]u(a,t)da=∫∞0nλ[(nλa)n−1e−nλa(n−1)!]u(a,t)da=nλIn(t). |
By introducing the sub-stage population densities described in Subsection 2.1.1:
Ij(t)=∫∞0(nλa)j−1e−nλa(j−1)!u(a,t)da,1≤j≤n, |
and
Mi(t)=∫∞0(mγξ)i−1e−mγξ(i−1)!v(ξ,t)dξ,1≤i≤m, |
the immature population size and the matured population size can be represented as
I(t)=n∑j=1Ij(t)andM(t)=m∑i=1Mi(t), respectively. |
Differentiating Ij(t) and Mi(t) and using (2.11) and (2.12), we can obtain a stage-structured model in ordinary differential equation form, which is same as system (2.10).
In this subsection, we assume the stage distribution follows a Dirac distribution. Similar to those in Subsection 2.1.2, we assume PM(ξ)≡1 for all M-stage ages ξ and that the distribution function for the immature stage has a mean duration τ. In this case,
v(0,t)=∫∞0[−P′I(a)]u(a,t)da=∫∞0δ(a−τ)u(a,t)da=u(τ,t) |
and the number of immature I(t) and mature M(t) individuals can be expressed as
I(t)=∫τ0u(a,t)daandM(t)=∫∞0v(ξ,t)dξ, |
respectively. Therefore, (2.11) gives
dI(t)dt=ddt(∫τ0u(a,t)da)=∫τ0(−∂u(a,t)∂a−μIu(a,t))da=−u(τ,t)+u(0,t)−μII(t). |
Similarly, we have the following equation for the matured population size M(t):
dM(t)dt=∫∞0(−∂v(ξ,t)∂ξ−μMv(ξ,t))dξ=u(τ,t)−μMM(t). |
It remains to find the maturation rate u(τ,t), which can be achieved by integration along characteristics. Let Vs(t)=u(t−s,t), then we have
ddtVs(t)=−μIVs(t) |
and Vs(t)=e−μI(t−t0)Vs(t0). If t≥τ, setting s=t−τ and t0=t−τ gives
u(τ,t)=Vt−τ(t)=e−μIτVt−τ(t−τ)=e−μIτu(0,t−τ)=B(M(t−τ))e−μIτ. |
If t<τ, let s=t−τ and t0=0, then
u(τ,t)=Vt−τ(t)=e−μItVt−τ(0)=e−μItu(τ−t,0). |
Therefore, the stage-structured population dynamics with a Dirac distribution for immature stage duration can be described by two sets of systems on different time intervals:
dI(t)dt=B(M(t))−u(τ−t,0)e−μIτ−μII(t)dM(t)dt=u(τ−t,0)e−μIτ−μMM(t)}for t∈[0,τ] |
and
dI(t)dt=B(M(t))−B(M(t−τ))e−μIτ−μII(t)dM(t)dt=B(M(t−τ))e−μIτ−μMM(t)} for t∈[τ,∞) | (2.13) |
It should be noted that the variable M(t) can be decoupled from the whole model system. Moreover, a scalar delay differential equation (the second equation of (2.13)) would be sufficient to reflect the long-term dynamics of the mature stage [35] under the conditions of a suitable initial value specified for M(θ) with θ∈[−τ,0].
In Subsections 2.1 and 2.2, two modeling approaches for physically structured population growth are presented in the form of an integral system (Eqs (2.1) and (2.2)) and a partial differential system (Eqs (2.11) and (2.12)). The relationship between the integral equation approach and the partial differential equation approach was established in [26,Chapter 13]. When the stage duration follows a gamma distribution, both modeling frameworks can be reduced into a system of ordinary differential equations, while a system of delay differential equations can be derived when the stage distribution follows a Dirac distribution. The reduction, without losing relevant growth information, makes it easier to investigate the population dynamics. The possibility of reducing a physiologically structured population model, such as those in Section 2.1, to an ordinary differential equation model has been investigated [36].
The integral system can be naturally formulated by applying ecological arguments for birth, stage progression and survival. Furthermore, the net reproduction of the population growth can be intuitively derived by using the integral equation nature of the system, with each term having clear biological interpretations. Moreover, the initial growth rate can be easily defined with the help of the linearized system, and its existence and uniqueness can be established through the use of simple mathematical arguments. As a byproduct, the important relationship between the net reproduction number R0 and the initial growth rate r0 can be easily established: the sign of R0−1 is the same as that of r0.
It should be noted that a generalized birth function B(M(t)) was assumed in the last two subsections, as it can easily accommodate the density-dependent self-regulation on the birth rate. However, the density-independence assumptions are imposed for the survivorship and stage-to-stage progression. When it is necessary to relax the density-independence assumptions on the stage-progression function PX(t) and survivorship ΠX(t) for the immature (X=I) and mature (X=M) stages, it may become challenging to propose appropriate probability functions. In this sense, the integral framework may not be a convenient way to describe the structured population size when more complicated density-dependent self-regulation is considered, as will be demonstrated by the model (2.14) presented later and those reviewed in the coming Section 3. Furthermore, an integral system can also be derived from a state-structured partial differential equation when the related survival and stage-progression functions can be formulated from the corresponding evolution system [37].
To conclude this section, we show that the age-structured partial differential system can be extended to accommodate the density-dependent survivorship due to intra specific competition when the gamma distribution is assumed. In this scenario, Eqs (2.11) and (2.12) can be revised to account for excess density-dependent mortality rate due to competition:
∂u(a,t)∂a+∂u(a,t)∂t=−μIu(a,t)−f(I(t))u(a,t),∂v(ξ,t)∂ξ+∂v(ξ,t)∂t=−μMv(ξ,t)−g(M(t))v(ξ,t),u(0,t)=B(M(t)), u(a,0)=u0(a), v(0,t)=nλIn(t) and v(ξ,0)=u0(ξ), | (2.14) |
where functions f(⋅) and g(⋅) represent the excess death rates due to intraspecific competition, dependent on total population sizes of the respective stage. Differentiating each sub-stage variable in Section 2.2.1 for immatures Ij(t), we have
dI1(t)dt=B(M(t))−(nλ+μI+f(I(t)))I1(t), |
and
dIj(t)dt=nλIj−1(t)−(nλ+μI+f(I(t)))Ij(t),1<j≤n. |
Similarly, for the sub-stages of mature individuals, we have
dM1(t)dt=−e−mγξv(ξ,t)|∞0−(mγ+μM+g(M(t)))∫∞0e−mγξv(ξ,t)dξ=nλIn(t)−(mγ+μM+g(M(t)))M1(t), |
and
dMi(t)dt=mγMi−1(t)−(mγ+μM+g(M(t)))Mi(t),1<i≤m. |
In summary, when intraspecific competition induces excess mortality in immatures and the immature stage duration follows a gamma distribution, an ordinary differential system can be reformulated from the age-structured partial differential equation modeling approach:
dI1(t)dt=B(M(t))−(nλ+μI+f(I(t)))I1(t),dIj(t)dt=nλIj−1(t)−(nλ+μI+f(I(t)))Ij(t),1<j≤n,dM1(t)dt=nλIn(t)−(mγ+μM+g(M(t)))M1(t),dMi(t)dt=mγMi−1(t)−(mγ+μM+g(M(t)))Mi(t),1<i≤m. | (2.15) |
This section is devoted to reviewing some population models with an assumed Dirac distribution for immature stage duration and intraspecific competition. In particular, we are interested in presenting different types of models that can be formulated under various assumptions on the effect of immature competition.
Considering the competition between immatures of the same age, Gourley and Liu [38] explored the following evolution equation for the population density u(a,t) for age a at time t
∂u(a,t)∂a+∂u(a,t)∂t=−μIu(a,t)−T(u(a,t)),0<a≤τ∂u(a,t)∂a+∂u(a,t)∂t=−μMu(a,t),a>τ. | (3.1) |
In this model, the competitive effects between immatures are given by a nonlinear function T(u(a,t)), which describes the influence of intraspecific competition among the immature individuals due to limited living space and resources. The competition among mature individuals is not taken into account.
By taking a similar arguments as those in Subsection 2.2.2, M(t)-equation can be written as
dM(t)dt=u(τ,t)−μMM(t). |
To close this equation, it is essential to obtain the explicit form of the maturation rate u(τ,t), which can be found by applying integration along characteristics. By introducing the function uξ(a)=u(a,a+ξ), the authors obtained
duξ(a)da=[∂u(a,t)∂a+∂u(a,t)∂t]t=a+ξ=[−μIu(a,t)−T(u(a,t))]t=a+ξ, |
which implies that
duξ(a)da=−μIuξ(a)−T(uξ(a)); |
hence,
∫uξ(a)uξ(0)dημIη+T(η)=−a. |
Here, uξ(0)=u(0,ξ)=B(M(ξ)). Choosing a=τ and ξ=t−τ, the maturation rate u(τ,t) at time t>τ can be solved explicitly from
∫B(M(t−τ))u(τ,t)dημIη+T(η)=τ. |
Since the function T: (0,∞)→R+ may be nonlinear, it is impossible to obtain an explicit expression u(τ,t)=Q(B(M(t−τ))) to illustrate the relationship between the maturation rate u(τ,t) at time t and the birth rate B(M(t−τ)) at time t−τ. However, this relationship y=Q(x) can be implicitly defined by
∫xQ(x)dημIη+T(η)=τ,y>0. | (3.2) |
Moreover, the function y=Q(x) is well-defined as T(⋅) is nonnegative and nondecreasing. Then
dM(t)dt=Q(B(M(t−τ)))−μMM(t). |
With this kind of competition in consideration, the authors of [38] show that all solutions are bounded for any birth function B(⋅). Linearizing the model at a boundary equilibrium gives verifiable and biologically interpretable conditions for its stability. In what follows, we will present several models of this type.
By specifying the nonlinear function as T(u(a,t))=βI(u(a,t))2 in (3.1), Liu, Röst and Gourley [39] investigated the following model
∂u(a,t)∂a+∂u(a,t)∂t=−μIu(a,t)−βI(u(a,t))2,0<a<τ | (3.3) |
where βI denotes the effect of intraspecific competition among immature individuals. In this case, it is possible to write down the function given by (3.2) explicitly. In fact, the new variable uξ(a)=u(a,a+ξ) satisfies
ddtuξ(a)=−μIuξ(a)−βI(uξ(a))2, |
which takes the form of a Bernoulli differential equation, with the solution explicitly given by
uξ(a)=μIuξ(0)e−μIaμI+βIuξ(0)(1−e−μIa). |
Therefore, when t>τ, setting a=τ and ξ=t−τ gives ut−τ(0)=u(0,t−τ)=B(M(t−τ)) and
u(τ,t)=Q(B(M(t−τ)))=μIB(M(t−τ))e−μIτμI+βIB(M(t−τ))(1−e−μIτ). |
With this special nonlinear function T(u(a,t))=βI(u(a,t))2, the long-term dynamics of M(t) can be described by the following delay differential equation:
dM(t)dt=μIB(M(t−τ))e−μIτμI+βIB(M(t−τ))(1−e−μIτ)−μMM(t). |
Arino, Wang and Wolkowicz [40] derived a model by applying a different approach with the aid of survival arguments for those being alive at time t−τ that is still alive at time t by using the following evolution equation
˜N′(t)=−μ˜N(t)−κ˜N2(t). |
By the technique of separation of variables and integration from t−τ to t, they obtained
˜N(t)=μ˜N(t−τ)μeμτ+κ(eμτ−1)˜N(t−τ). |
Putting this density-dependent term into a logistic equation with a birth rate γ, the authors formulated an alternative logistic delay differential equation with a time delay τ:
N′(t)=γμN(t−τ)μeμτ+κ(eμτ−1)N(t−τ)−μN(t)−κN2(t). | (3.4) |
It is shown that the population dies out when the delay is too large. The existence of a positive equilibrium, and its relationship with parameter values are further illustrated in [40].
Using a similar argument as that in [40], Lin, Wang and Wolkowicz [41] formulated a logistic equation with distributed delays. The time delay is distributed according to a kernel function k(s) by using a mean delay τ, that is:
k(s)≥0,∫∞0k(s)ds=1,∫∞0sk(s)ds=τ. |
Then the discrete delay logistic-type equation (3.4) can be extended to the following one:
N′(t)=γ∫∞0μe−μsN(t−s)k(s)μ+κ(1−e−μs)N(t−s)ds−μN(t)−κN2(t). | (3.5) |
The delay kernel can take a variety of functions, such as the Dirac delta function, a uniform distribution, gamma distribution and tent distribution. When the kernel function is Dirac delta function, Eq (3.5) is exactly the discrete delay case described by Eq (3.4). A threshold result for survival and extinction was established in [41]: the global attractivity of the unique positive equilibrium and the zero equilibrium are established under different parameter regimes.
Considering the excess mortality due to intraspecific competition between individuals at the same life stage, Fang, Gourley and Lou [42] assumed the Dirac distribution described by immatures for the model (2.14), copied as follows for easy reference:
∂u(a,t)∂a+∂u(a,t)∂t=−μIu(a,t)−f(I(t))u(a,t),a≤τ∂u(a,t)∂a+∂u(a,t)∂t=−μMu(a,t)−g(M(t))u(a,t),a>τu(0,t)=B(M(t)) and u(a,0)=u0(a). |
In this case, the number of individuals in each stage are
I(t)=∫τ0u(a,t)da and M(t)=∫∞τu(a,t)da. |
Differential equations for two variables I(t) and M(t) when t>τ can be derived as
dI(t)dt=−u(τ,t)+u(0,t)−μII(t)−f(I(t))I(t),dM(t)dt=−u(∞,t)+u(τ,t)−μMM(t)−g(M(t))M(t), | (3.6) |
where u(0,t)=B(M(t)) and u(∞,t)=0. The maturation rate u(τ,t) can be explicitly solved by integration along characteristics, as follows:
u(τ,t)=Vt−τ(t)=B(M(t−τ))e−μIτ−∫τ0f(I(t−τ+ξ))dξ for t>τ. |
It should be highlighted that (3.6) explicitly couples both variables I(t) and M(t) together, which is different from the previous scalar delay differential equations for the population size at the mature stage (such as those in Subsections 2.1.2 and 2.2.2, and Subsection 3.1). Furthermore, since the maturation rate u(τ,t) is a decreasing function of I as the function f(⋅) is assumed to be increasing, there are novel challenges in the theoretical analysis. In particular, the stability analysis of the equilibria becomes difficult due to the strong coupling of two state variables. A generic convergence result is established for small delays by using monotone dynamical systems theory and exponential ordering [42].
Another larval competition model was proposed and studied by Liu, Röst and Gourley [39], which follows
∂u(a,t)∂a+∂u(a,t)∂t=−μIu(a,t)−ϵu(a,t)∫τ0p(¯a,a)u(¯a,t)d¯a,0<a<τ. | (3.7) |
In this model, ϵ characterizes the intensity of population competition among the immature individuals, and p(¯a,a) is an adjustable parameter to describe various competition types: (ⅰ) p(¯a,a) being a constant if an immature individual is likely to compete with all other immature individuals with the same competitive pressure, regardless of age; (ⅱ) p(¯a,a)=0 as ¯a<a, implying that an immature individual only competes with older individuals; (ⅲ) p(¯a,a)=δ(¯a−a) with a Dirac delta function δ(⋅) if competition occurs among individuals of the same age, which was considered in (3.3).
To transform the model into an ordinary differential equation form with time delay, it is essential to find the maturation rate u(τ,t) by using the evolution of immature population density described by (3.7). It seems impossible to obtain explicit solutions for general cases, and the authors in [39] applied perturbation theory to seek the solution of the following two specific forms:
u(a,t)=u0(a,t)+ϵu1(a,t)+O(ϵ2) and u(a,t)=u0(a,t)exp(−ϵu1(a,t)+O(ϵ2)), |
with u0(0,t)=B(M(t)) and u1(0,t)=0. Integrating (3.7) along characteristics gives the maturation rate u(τ,t) when t>τ, and two alternative models for the mature population M(t) are given by
dM(t)dt=−μMM(t)+B(M(t−τ))e−μIτ[1−ϵ∫τ0∫τ0p(¯a,s)B(M(s+t−τ−¯a))e−μI¯ad¯ads] |
and
dM(t)dt=−μMM(t)+B(M(t−τ))e−μIτexp(−ϵ∫τ0∫τ0p(¯a,s)B(M(s+t−τ−¯a))e−μI¯ad¯ads). |
This model, proposed under the simple assumption that an individual larva experiences competition from other larvae during development, poses rich dynamics. In particular, the existence of multiple co-existing equilibria is shown in some parameter regimes.
When the duration of staying in the immature stage is regulated by the population density, it would be more convenient to use another variable x called "state" [30], which generalizes the concept of the age, to describe the population density evolution. Based on the fact that maturation can be measured to some extent by state, the maturity of an individual occurs when its state x achieves a fixed threshold l. Let u(x,t) represent the population density of immature individuals of state x at time t, then the immature population size I(t) at time t counts all individuals with a state variable x smaller than l, that is
I(t)=∫l0u(x,t)dx. |
This new variable makes it possible to describe the case that the rate of change of the state x with respect to time is not constant, but is dependent on the population density (see Eq (3.8)).
Assuming all individuals at the immature stage compete for limited resources, which slows their development, Gourley, Liu and Lou [43] used the following equation to describe the rate of change for the length variable x at time t:
dxdt=P(t,I(t)), | (3.8) |
which relies both on time t and on the total number of individuals comprising the immature population I(t). Here, the function P(t,I) is decreasing on the variable I and is dependent on time t to reflect the time-changing environmental impacts on development.
To derive the partial differential equation for u(x,t) to obtain that in (3.1), the authors used the following argument: After a period of δt, an immature individual will have developed a length of δx, namely
u(x+δx,t+δt)=u(x,t)−μIu(x,t)δt, |
which implies that
∂u(x,t)∂t+P(t,I(t))∂u(x,t)∂x=−μIu(x,t), x≤l | (3.9) |
by a Taylor expansion. Taking the derivative of I(t) and using Eq (3.9), one obtains
dI(t)dt=−μII(t)+P(t,I(t))(i(0,t)−i(l,t)), |
where P(t,I(t))i(0,t) denotes the birth rate, that is P(t,I(t))i(0,t)=B(M(t)), and P(t,I(t))i(l,t) denotes the maturation rate, which will be calculated according to the birth rate at time t−τ(t). The term u(x,t) relies on whether (x,t) is above or below the characteristic x=X(t), where
X(t)=∫t0P(ξ,I(ξ))dξ. |
Introducing a parameter s such that dtds=1, then dxds=P(t,I(t)). The parameter s is used to describe the position along a particular characteristic and s=0 corresponds to a boundary.
When x≥X(t), a characteristic (x(s),t(s)) meets the x-axis, which implies that t=0 when s=0. Setting t(0)=0 gives x−x(0)=X(t). It follows from (3.9) that
ddsu(x(s),t(s))=−μIu(x(s),t(s)), |
which implies that
u(x(s),t(s))=u(x(0),t(0))e−μIs; |
thus
u(x,t)=u(x−X(t),0)e−μIt,x≥X(t). |
When x≤X(t), a characteristic (x(s),t(s)) meets the x-axis, which implies that x(0)=0 and t=s+t(0). The corresponding s-value for a particular point (x,t) can be defined by
x=∫tt(0)P(ξ,I(ξ))dξ=∫tt−sP(ξ,I(ξ))dξ. |
Defining L(x,t) to be the root s, which implies that
∫tt−L(x,t)P(ξ,I(ξ))dξ=x. |
Therefore,
u(x,t)=i(0,t−L(x,t))e−μIL(x,t)=B(M(t−L(x,t)))P(t−L(x,t),I(t−L(x,t)))e−μIL(x,t),x≤X(t). |
Hence,
i(l,t)=i(0,t−L(l,t))e−μIL(l,t)=B(M(t−L(l,t)))P(t−L(l,t),I(t−L(l,t)))e−μIL(l,t), |
and the corresponding maturation delay τ(t)=L(l,t) for individuals developing to the mature stage at time t depends on the immature population size I(t) as specified by
∫tt−τ(t)P(ξ,I(ξ))dξ=l. |
The last integral-algebraic equation has a clear biological interpretation: an individual that develops to the mature stage at time t should be born at t−τ(t) such that the accumulative length increase during the time interval [t−τ(t),t] attains the critical value l. We should mention that similar integral forms to characterize the density-dependent time delay can also be found in other earlier studies, such as [44,45,46].
Based on the fact that the variable x remains valid for the immature population, the equation describing the number of individuals at the mature stage can be expressed as
dM(t)dt=−μMM(t)+maturation rate=−μMM(t)+P(t,I(t))B(M(t−τ(t)))P(t−τ(t),I(t−τ(t)))e−μIτ(t). |
In addition, the size of the immature population I(t) satisfies
dI(t)dt=−μII(t)+B(M(t))−P(t,I(t))B(M(t−τ(t)))P(t−τ(t),I(t−τ(t)))e−μIτ(t). |
Results on the boundedness of solutions and the linear stability of the equilibria are presented in [43]. The boundedness of solutions holds even for unbounded birth functions within certain conditions. It is also shown that if an equilibrium is locally stable in the absence of competition among larvae, then the equilibrium is stable in the presence of weak competition.
Considering extreme cases that the development may be paused due to immature competition, termed as diapause, Brunner, Gourley, Liu and Xiao [47] studied the following size growth rate function
P(I)={P0,I≤Ic,0,I>Ic, |
with the constant P0. This form implies that the immature individuals develop at a constant rate P0 when their total number is less than Ic, while the growth of the immature population is paused due to high competition pressure when its size exceeds Ic. Therefore, the change of an immature individual's size can be described as
dxdt=P(I(t)), |
and the growth rate function is dependent on the immature population size I(t).
Since the occurrence of diapause may increase the maturation time τ(t) needed, τ(t)≥l/P0 with l being the critical size at maturity. As a matter of fact, when P(⋅) is nonnegative, but not strictly positive everywhere, τ(t) can be defined as
τ(t)=inf{s>0:∫st−sP(I(ξ))dξ=l}, |
which reduces to
∫tt−τ(t)P(I(ξ))dξ=l |
if P(⋅) is strictly positive. Based on the integration along characteristics, the delay differential system can be formulated as follows:
dI(t)dt=−μII(t)+B(M(t))−B(M(t−τ(t)))P(I(t))P(I(t−τ(t)))e−μIτ(t),dM(t)dt=−μMM(t)+B(M(t−τ(t)))P(I(t))P(I(t−τ(t)))e−μIτ(t). |
When an Allee effect is assumed for the birth rate function B(⋅), diapause may induce population extinction even for large initial population sizes. Diapause may also introduce periodic solutions that can arise even for a strictly increasing birth function.
Development from one life stage to the next takes time while the time spent in each stage may be synchronized or vary between individuals, giving rise to various distributions of development times in each stage for different species. These distributions intuitively can play important roles in the transition rates among different life stages. In this manuscript, two basic modeling frameworks to describe demographic changes of population dynamics, based on integral and partial differential systems, are presented. These models can be reduced to ordinary and delay differential stage-structured models under gamma and Dirac distribution assumptions. It is evident that each framework has its advantages and inherent limitations. In particular, the integral system can be naturally formulated by checking the stage progression of individuals. Furthermore, the net reproduction number and initial growth rate can be explicitly derived from the integral system. However, it becomes challenging to integrate the density-dependent regulations on the stage distribution and survival probabilities in an integral system due to difficulties in formulating appropriate survival probability functions and stage duration distributions. This may be suitably resolved by using structured partial differential equation models. By applying further assumptions to these density-dependent regulations, the partial differential system can be reduced to different forms, and in particular, various delay differential equation models were reviewed in this study.
When the impact of density regulation on immature individual survival and development is negligible, it is evident from Section 2 that the equation for the matured population size is decoupled from the integral system ((2.1) and (2.2)) as the variable accounting for the immature population size does not appear in (2.2). A similar observation can be made regarding the delay differential equation model in Sections 2.1.2 and 2.2.2 when the Dirac distribution is assumed for the stage duration of immatures. From an analytical point of view, this observation makes it possible to analyze the dynamics of the mature stage M(t) first, and then to feed the equation of the immature stage with the dynamic profile M(t). Since the extinction and persistence of the species can be predicted from those of each stage, it would be sufficient to show the extinction/persistence of mature individuals from the decoupled equation for M(t), as analyzed in [42] for the case when the immature competition force f(⋅)=0. When a gamma distribution is assumed for the stage duration of immatures, the maturation rate becomes nλIn(t) in the system of ordinary differential equations described in Sections 2.1.1 and 2.2.1, which makes it impossible to decouple the variables for mature stages from the whole system at first glance. However, if one revisits the definition of In(t) in (2.9), it can be expressed in terms of M(t) with a distributed delay kernel.
Furthermore, a Dirac distribution with an average duration τ can be approximated by a gamma distribution as shown in (2.7), with γ=1/τ and a large n (such that the variance of the gamma distribution τ2/n is very small), as shown in Figure 1. Intuitively, the delay differential equation model under Dirac distribution assumption would also be approximated by n ordinary differential equations under the gamma distribution assumption for a large n. As a matter of fact, this can be rigorously shown by the linear chain trick [48], as illustrated in Section 2.2.2 by observing the maturation rate when t>τ is
v(0,t)=u(τ,t)=B(M(t−τ))e−μIτ=∫∞0δ(a−τ)e−μIaB(M(t−a))da≈∫∞0[−P′I(a)]e−μIaB(M(t−a))da. |
In the above expression, δ(⋅) is the Dirac-delta function and PI(a) takes the gamma distribution as in (2.7).
The stage-structured modeling idea in this manuscript can easily accommodate spatial movements of individuals. In particular, when individuals are performing random movements, a reaction-diffusion model with/without time delay can be formulated when the exponential and Dirac distributions are assumed for the stage duration [20,49,50]. In particular, a nonlocal delay term can be formulated when the Dirac distribution is assumed and immature individuals move during development. Interested readers may refer to the pioneering model formulations by Stephen Gourley and his collaborators, such as those in [50,51,52,53].
Other modeling frameworks, such as matrix population models and individual-based models, are also important tools to incorporate the variation of individual-level demographic characteristics, which are beyond the scope of this review. We refer the interested readers to [1,2] for incorporating stage duration distributions in other model forms, such as matrix models and statistical stage-duration distribution models. Further biotic and abiotic factors may also impact the stage duration distributions: for example, the seasonal environmental oscillations can induce seasonal developmental delays and seasonal diapauses, which have been modeled in [54,55]. We leave these topics for further investigation.
We are very grateful to the anonymous referees for their careful reading and valuable comments, as well as some inspiring references.
The authors declare there is no conflict of interest.
[1] |
C. D. Huang, L. M. Cai, J. D. Cao, Linear control for synchronization of a fractional-order time-delayed chaotic financial system, Chaos Solit. Frac., 113 (2018), 326-332. doi: 10.1016/j.chaos.2018.05.022
![]() |
[2] |
Q. Gao, J. H. Ma, Chaos and Hopf bifurcation of a finance system, Nonlinear Dyn., 58 (2009), 209-216. doi: 10.1007/s11071-009-9472-5
![]() |
[3] |
X. L. Gong, X. Liu, X. Xiong, Chaotic analysis and adaptive synchronization for a class of fractional order financial system, Phys. A: SMA., 522 (2019), 33-42. doi: 10.1016/j.physa.2019.01.138
![]() |
[4] | M. C. Zhao, J. W. Wang, H∞ control of a chaotic finance system in the presence of external disturbance and input time-delay, Appl. Math. Comput., 233 (2014), 320-327. |
[5] | Y. Cao, Chaotic synchronization based on fractional order calculus financial system, Chaos Solit. Frac., 130 (2020), 1-7. |
[6] | A. Hajipour, H. Tavakoli, Dynamic Analysis and Adaptive Sliding Mode Controller for a Chaotic Fractional Incommensurate Order Financial System, Int. J. Bifur. Chaos, 27 (2017), 1-14. |
[7] |
R. F. Rao, S. M. Zhong, Impulsive control on delayed feedback chaotic financial system with Markovian jumping, Adv. Diff. Equ., 2020 (2020), 1-18. doi: 10.1186/s13662-019-2438-0
![]() |
[8] | R. F. Rao, S. M. Zhong, Input-to-state stability and no-inputs stabilization of delayed feedback chaotic financial system involved in open and closed economy, Disc. Contin. Dyn. Sys., Ser. S., 2020 (2020), 1-19. |
[9] | R. F. Rao, Global stability of a markovian jumping chaotic financial system with partially unknown transition rates under impulsive control involved in the positive interest rate, Mathematics, 7 (2019), 1-15. |
[10] |
J. Duan, Financial system modeling using deep neural networks (DNNs) for effective risk assessment and prediction, J. Franklin Inst., 356 (2019), 4716-4731. doi: 10.1016/j.jfranklin.2019.01.046
![]() |
[11] |
Q. K. Song, J. D. Cao, Dynamical behaviors of discrete-time fuzzy cellular neural networks with variable delays and impulses, J. Franklin Inst., 345 (2008), 39-59. doi: 10.1016/j.jfranklin.2007.06.001
![]() |
[12] |
D. H. Zheng, H. B. Zhang, J. A. Zhang, W. X. Zheng, S. W. Su, Stability of asynchronous switched systems with sequence-based average dwell time approaches, J. Franklin Inst., 357 (2020), 2149-2166. doi: 10.1016/j.jfranklin.2019.11.067
![]() |
[13] | R. Sakthivel, S. Selvi, K. Mathiyalagan, P. Shi, Reliable mixed H∞ and passivity-based control for fuzzy markovian switching systems with probabilistic time delays and actuator failures, IEEE Trans. Cyb., 23 (2015), 2720-2731. |
[14] | T. Takagi, M. Sugeno, Fuzzy identification of systems and its application to modeling and control, IEEE Trans. Syst., 15 (1985), 116-132. |
[15] |
Q. K. Song, Z. J. Zhao, J. X. Yang, Passivity and passification for stochastic Takagi-Sugeno fuzzy systems with mixed time-varying delays, Neurocomputing, 122 (2013), 330-337. doi: 10.1016/j.neucom.2013.06.018
![]() |
[16] |
Z. P. Wang, H. N. Wu, Fuzzy impulsive control for uncertain nonlinear systems with guaranteed cost, Fuzzy Sets Syst., 302 (2016), 143-162. doi: 10.1016/j.fss.2015.09.026
![]() |
[17] |
Q. K. Song, Z. D. Wang, Dynamical behaviors of fuzzy reaction-diffusion periodic cellular neural networks with variable coefficients and delays, Appl. Math. Mod., 33 (2009), 3533-3545. doi: 10.1016/j.apm.2008.11.017
![]() |
[18] |
S. J. Long, Q. K. Song, X. H. Wang, D. S. Li, Stability analysis of fuzzy cellular neural networks with time delay in the leakage term and impulsive perturbations, J. Franklin Inst., 349 (2012), 2461-2479. doi: 10.1016/j.jfranklin.2012.05.009
![]() |
[19] |
D. Yang, X. D. Li, S. J. Song, Design of state-dependent switching laws for stability of switched stochastic neural networks with time-delays, IEEE Trans. Neu. Net. Learn. Sys., 31 (2020), 1808-1819. doi: 10.1109/TNNLS.2019.2927161
![]() |
[20] |
J. Cheng, J. H. Park, Y. J. Liu, Z. J. Liu, L. M. Tang, Finite-time H∞ fuzzy control of nonlinear Markovian jump delayed systems with partly uncertain transition descriptions, Fuzzy Sets Syst., 314 (2017), 99-115. doi: 10.1016/j.fss.2016.06.007
![]() |
[21] |
J. J. Ren, X. Z. Liu, H. Zhu, S. M. Zhong, C. Wu, Exponential H∞ synchronization of switching fuzzy systems with time-varying delay and impulses, Fuzzy Sets Syst., 365 (2019), 116-139. doi: 10.1016/j.fss.2018.05.019
![]() |
[22] | Q. Li, X. Z. Liu, Q. X. Zhu, S. M. Zhong, D. Zhang, Distributed state estimation for stochastic discrete-time sensor networks with redundant channels, Appl. Math. Comput., 343 (2019), 230-246. |
[23] | S. Y. Dong, H. Zhu, S. M. Zhong, K. B. Shi, J. Cheng, W. Kang, New result on reliable H∞ performance state estimation for memory static neural networks with stochastic sampled-data communication, Appl. Math. Comput., 364 (2020), 1-17. |
[24] |
M. Z. Luo, X. Z. Liu, S. M. Zhong, J. Cheng, Synchronization of multi-stochastic-link complex networks via aperiodically intermittent control with two different switched periods, Phys. A: SMA., 509 (2018), 20-38. doi: 10.1016/j.physa.2018.05.145
![]() |
[25] |
X. D. Li, X. L. Fu, Stability analysis of stochastic functional differential equations with infinite delay and its application to recurrent neural networks, J. Comput. Appl. Math., 234 (2010), 407-417. doi: 10.1016/j.cam.2009.12.033
![]() |
[26] |
R. F. Rao, S. M. Zhong, Z. L. Pu, Fixed point and p-stability of T-S fuzzy impulsive reaction-diffusion dynamic neural networks with distributed delay via Laplacian semigroup, Neurocomputing, 335 (2019), 170-184. doi: 10.1016/j.neucom.2019.01.051
![]() |
[27] |
H. Cheng, S. M. Zhong, X. Q. Li, Q. S. Zhong, J. Cheng, Exponential synchronization of delayed memristor-based neural networks with stochastic perturbation via nonlinear control, Neurocomputing, 340 (2019), 90-98. doi: 10.1016/j.neucom.2019.02.032
![]() |
[28] | D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes Math., 840, Berlin, New York : Springer-Verlag, 1981. |
[29] |
D. Horstmann, M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Diff. Equ., 215 (2005), 52-107. doi: 10.1016/j.jde.2004.10.022
![]() |
[30] |
R. F. Rao, Delay-dependent exponential stability for nonlinear reaction-diffusion uncertain cohen-grossberg neural networks with partially known transition rates via Hardy-Poincare inequality, Chin. Ann. Math., Ser. S., 35 (2014), 575-598. doi: 10.1007/s11401-014-0839-7
![]() |
[31] |
R. F. Rao, S. M. Zhong, X. R. Wang, Stochastic stability criteria with LMI conditions for Markovian jumping impulsive BAM neural networks with mode-dependent time-varying delays and nonlinear reaction-diffusion, Commun. Nonlinear Sci. Num. Simu., 19 (2014), 258-273. doi: 10.1016/j.cnsns.2013.05.024
![]() |
[32] | R. F. Rao, Stability analysis of nontrivial stationary solution and constant equilibrium point of reaction-diffusion neural networks with time delays under Dirichlet zero boundary value, Preprint, 2020, 2020040277 (doi: 10.20944/preprints202004.0277.v6).Availablefrom:https://www.preprints.org/manuscript/202004.0277/v6 |
[33] | X. R. Mao, Stochastic Differential Equations and Applications, Horwood, 1997. |
[34] |
D. Xu, Z. G. Yang, Y. M. Huang, Existence-uniqueness and continuation theorems for stochastic functional differential equations, J. Diff. Equ., 245 (2008), 1681-1703. doi: 10.1016/j.jde.2008.03.029
![]() |
[35] |
W. J. Xie, Q. X. Zhu, Input-to-state stability of stochastic nonlinear fuzzy Cohen-Grossberg neural networks with the event-triggered control, Int. J. Control, 93 (2020), 2043-2052. doi: 10.1080/00207179.2018.1540887
![]() |
[36] |
X. D. Li, X. L. Zhang, S. J. Song, Effect of delayed impulses on input-to-state stability of nonlinear systems, Automatica, 76 (2017), 378-382. doi: 10.1016/j.automatica.2016.08.009
![]() |
[37] | X. D. Li, X. Y. Yang, J. D. Cao, Event-triggered impulsive control for nonlinear delay systems, Automatica, 117 (2020), 108981. |
[38] |
P. Li, X. D. Li, Input-to-state stability of nonlinear impulsive systems via Lyapunov method involving indefinite derivative, Math. Comp. Simu., 155 (2019), 314-323. doi: 10.1016/j.matcom.2018.06.010
![]() |
[39] | H. T. Zhu, P. Li, X. D. Li, H. Akca, Input-to-state stability for impulsive switched systems with incommensurate impulsive switching signals, Commun. Nonlinear Sci. Num. Simu., 80 (2020), 104969. |
[40] |
X. D. Li, X. Y. Yang, S. J. Song, Lyapunov conditions for finite-time stability of time-varying time-delay systems, Automatica, 103 (2019), 135-140. doi: 10.1016/j.automatica.2019.01.031
![]() |
[41] | X. D. Li, X. Y. Yang, T. W. Huang, Persistence of delayed cooperative models: Impulsive control method, Appl. Math. Comput., 342 (2019), 130-146. |
[42] | X. D. Li, J. H. Shen, R. Rakkiyappan, Persistent impulsive effects on stability of functional differential equations with finite or infinite delay, Appl. Math. Comput., 329 (2018), 14-22. |
[43] |
D. Yang, X. D. Li, J. H. Shen, Z. J. Zhou, State-dependent switching control of delayed switched systems with stable and unstable modes, Math. Meth. Appl. Sci., 41 (2018), 6968-6983. doi: 10.1002/mma.5209
![]() |
[44] |
P. Li, X. D. Li, Input-to-state stability of nonlinear impulsive systems via Lyapunov method involving indefinite derivative, Math. Comput. Simu., 155 (2019), 314-323. doi: 10.1016/j.matcom.2018.06.010
![]() |
[45] |
W. C. Chen, Dynamics and control of a financial system with time-delayed feedbacks, Chaos Soli. Frac., 37 (2008), 1198-1207. doi: 10.1016/j.chaos.2006.10.016
![]() |
[46] | J. H. Ma, Y. S. Chen, Study for the bifurcation topological structure and the global complicated character of a kind of nonlinear finance system (I), Appl. Math. Mech., 11 (2001), 1240-1251. |
[47] | J. H. Ma, Y. S. Chen, Study for the bifurcation topological structure and the global complicated character of a kind of nonlinear finance system (II), Appl. Math. Mech., 12 (2001), 1375-1382. |
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