Kohyama, Takashi

Nakajima, Hisao

Takada, Takenori

Kyoto University Research Information Repository

TitleA mathematical analysis in discrete transition system (Theoryof Biomathematics and its Applications IV)Author(s)Takada, Takenori; Kohyama, Takashi; Nakajima, HisaoCitation数理解析研究所講究録 (2008), 1597: 111-117Issue Date2008-05URL http://hdl.handle.net/2433/81743RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityA mathematical analysis in discrete transition system.Hokkaido University Takenori TakadaHokkaido University Takashi KohyamaRitusmeikan University Hisao NakajimaIntroductionIt is well known that forest community is consisted of multiple layers of tree populations,from shrubs to taller tree species; especially, tropical forests have 3 or 4 layers with 30 to 50meters tall. Individuals of the same species also occupy the different layers as they grow upfrom seedlings or saplings to mature trees. They transit from lower layer to middle and toplayer during their lifetime. Therefore, dynamics of multi-layer population can be described asthe transitions among discrete layers, using variables which represent the number ofindividuals at each layer. In this paper, we constructed discrete transition system andexamined the local stability of the equilibrium of the system. The preceding studies examinedthe local stability of the two-layer system and obtained several theorems on permanence andpersistence (Thieme (2003), Baer et a1.(2006)). We extended the system to three-layered oneand discussed about the relationship between local stability and the pattem of interactionamong individuals at different layers. We showed that one-sided interaction (or $asymmeu\cdot ic$competition) through light resource promotes the stability of tree populations.Two-layer transition systemA species is distributed both in the lower and upper layer and the number of individuals atlower and upper layer are denoted as $X_{1}$ and $x_{2}$ , respectively. The new-bom individuals arerecruited into only the lower layer with fecundity function $f(x_{1}.x_{2})$ , and individuals step upfrom the lower to upper layer when they grow up with growth rate function $g(x_{1},x_{2})$ . Onlythe upper-layer individuals reproduce the new-born individuals. The mortality rate functionsof lower- and upper-layer individuals are denoted as $m_{1}(x_{1},x_{2})$ and $m_{2}(x_{1},x_{2})$ ,respectively. Then, discrete transition system can be written as:1597 2008 111-117 111$\{\begin{array}{l}\frac{dx_{1}}{dt}=f(x_{1},x_{2})x_{2}-g(x_{1},x_{2})x_{1}-m_{1}(x_{1},x_{2})x_{1}\frac{d\kappa_{2}}{dt}=g(x_{1},x_{2})x_{1}-m_{2}(x_{1},x_{2})x_{2}\end{array}$ (1)where the first derivatives of $f(x_{1},x_{2})$ and $g(x_{1},x_{2})$ with respect to $x_{j}$ are non-positive andthat of $m_{i}(x_{1},x_{2})$ is non-negative because of the negative density effect.Therefore, the system has always $(0,0)$ and positive equilibria as long as the density effect isoperated, and the number of positive equilibria depends on the functional forms of vital ratefunctions. Zero equilibrium is unstable, as far as$f(0,0) \frac{g(0,0)}{g(0,0)+m_{1}(0,0)}>m_{2}(0,0)$ . (2)Under the condition (2), we examined the local stability of the positive equilibrium withoutspecifying the functional forms of vital rates. The biological constraints of those functionsare:$f(x_{1}.x_{2}),g(x_{1},x_{2}),m_{j}(x_{1},x_{2})>0$ and $\frac{f}{fx_{j}}\leq 0,\frac{\partial g}{\partial x_{j}}\leq 0,\frac{\partial m_{i}}{\partial x_{j}}\geq 0$ . (3)(i) The mode of density-dependenceThe modes of density-dependence, i.e. interaction among individuals, are also defined, usingthe first derivative of a function with respect to stage variables. Suppose that mortality ratefunction of lower-layer individuals depends negatively on only the number of upper-layerindividuals, which means one-sided interaction by taller individuals. Then,$\frac{\partial m_{1}}{\partial x_{2}}>0$ and $\frac{\partial m_{1}}{\partial x_{1}}=0$ (One-sided interaction).On the other hand, mortality rate function of upper-layer individuals depends negatively ononly the number of lower-layer individuals, which means inverse one-sided interaction bylower individuals (we, hereafter, call it “revolution”).Then,$\frac{\partial m_{2}}{\partial x_{1}}>0$ and $\frac{\partial m_{2}}{\partial x_{2}}=0$ (Revolution).If a function depends both on lower- and upper-layer individuals, it is called both-sidedinteraction as:112$\frac{\partial m_{2}}{\partial x_{1}}>0$ and $\frac{\partial m_{2}}{\partial x_{2}}>0$ (Both-sided interaction).(ii) Single density effectWe firstly examine the single density effect on the local stability of positive equilibrium. Ifonly a vital rate function $(h)$ depends on only one variable, we call it single’ density effect. Italso means other vital rate functions are constant and density-independent. There are fourvital rate functions and two variables in two-layer transition system. Therefore, we analyzedeight kinds of the systems with single density effect (Table 1).We constructed the Jacobian matrix at the positive equilibrium to examine the localstability and evaluated the signs of the trace and the determinant, using the signs of vital ratefunctions (positive) and the derivatives (zero when it is constant or negative when it is densitydependent). If the trace is always negative and the determinant is always positive, the positiveequilibrium is always stable and never unstable irrespective of the specific functional forms ofvital rate functions. If the sign of either the trace or the determinant is not definite, it meansthe equilibrium could be unstable depending on the strength of the density-dependence of thefunction in question or the parameters.In Table 1, most of the single density effects lead to the $s$tability of positiveTable 1 The result of single density effect for local stability of positive equilibrium.‘Stable” in the cells means the positive equilibrium is never unstable irrespective of thestrength of density-dependence and “Unstable” does the equilibrium could be unstabledepending on the strength of density-dependence.113equilibrium. Only the density effect by the number of lower individuals on the growth ratefunction could be the cause of the instability.(iii) Multiple density effectsIn general, the system (1) has several density effects at the same time. In somecases, the fecundity function depends both on $x_{1}$ and $x_{2}$ . In other cases, the fecundity functiondepends on $x_{1}$ and the growth rate function depends on $x_{2}$ . We analyzed the loca] stabilityanalysis in such multiple density effects. The number of the combinations ofdensity-dependence of the vital rates functions is $2^{8}-1$ (Table 2). We constructed a computerprogram to examine the local stability of the positive equilibrium. In the program, weevaluated the signs of the trace and the determinant of the Jacobian, only using the signs ofTable 2 The result of multiple density effects for local stability of positive equilibrium.“Yes” in the cells represents the density dependence by a variable in question. The hyphenrepresents no density-dependent effect. $*$ represents the wild card of either‘Yes” orhyphen.114vital rate functions and the derivatives.In Table 2, the result of single density effect is also represented in the former part ofthe table, which shows the same result as in Table 1. In the cases of multiple density effects,two types of instability could occur. One is the case where the density dependence ofgrowth rate function by $x_{1}$ (revolution factor). The other cause is the density dependence ofmortality rate function by $x_{1}$ (revolution factor) when coupled wiith $\frac{\partial f}{\partial x_{2}}<0$ or $\frac{\partial m_{1}}{\partial x_{2}}>0$ .Therefore, if two revolution factors are removed, the system is always stable. Summanizingthe result of two-layer system, the positive equilibrium is never unstable if there is no“revolution”.$Thre\epsilon\cdot 1ay_{\bm{t}}r$ transition systemThe same analysis was conducted in three-layer transition system. The discrete transitionsystem can be written as:$\{\begin{array}{ll}\frac{dx_{1}}{dt}=f(x)x_{3}-g_{1}(x)x_{1}-m_{1}(x)x_{1} \frac{dx_{2}}{dt}=g_{I}(x)x_{I}-g_{2}(x)x_{2}-m_{2}(x)x_{2} \bm{x}=(x_{1},x_{2},x_{3}).\frac{dx_{3}}{dt}=g_{2}(x)x_{2}-m_{3}(x)x_{3} \end{array}$ (4)Therefore, the system has always zero and positive equilibria as long as the density effect isoperated, and the number of positive equilibria depends on the functional forms of vital ratefunction$s$ . Under the condition (2), we examined the local stability of the positive equilibriumwithout specifying the functional form$s$ of vital rates.(i) Single density effectWe firstly examine the single density effect on the local stability of positive equilibrium.There are six vital rate functions and three variables in three-layer tran$s$ ition system.Therefore, we analyzed local stabilities of 18 kinds of the systems with single density effect(Table 3).In Table 3, 6 kinds of single density effects lead to the instability of positiveequilibrium. Four kinds of single density effects are‘Revolution” type and the other two aoe“one-sided interaction” type.(ii) Multiple density effects115We also analyzed the local stability analysis in the multiple density effects. The number of theTable 3 The result of single density effects in three-layer transition system.combinations of density-dependence of the vital rates functions is $2^{18}-1$ $(262,143)$. Weexamined the local stability of the positive equilibrium, using the program to evaluate thesigns of the trace and the determinant. The result clarifies three types of causes for instability.The first cause is a single density effect that leads to the instability. If the combination ofdensity effect includes the single density effect, it always causes the instability. The secondis a density effect on growth rate at middle layer by lower individuals. The density effectcould be the cause of instability when coupled with one or more of the followingdensity-dependence:$\frac{\partial f}{\partial x_{2}}<0,\frac{\partial m_{1}}{\partial x_{2}}>0,\frac{\partial g_{1}}{\partial x_{2}}<0,\frac{\partial g_{1}}{\partial x_{3}}<0,\frac{\partial m_{2}}{\partial x_{3}}>0,\frac{\partial g_{2}}{\partial x_{3}}<0$ (5).The third is a density effect on mortality rate at upper layer by lower individuals. The densityeffect could be the cause of instability when coupled with one or more of the inequalities (5).Summarizing the result in three-layer system, the positive equilibrium could be unstable evenif there is no ‘revolution”.References116Baer, S.M., Kooi, B. W. , Kuznetsov, YU. A. and Thieme, H. R. (2006) Multiparametricbifurcation analysis of a basic two-stage population model. SIAM J Appl. Math 66:1339-1365.Thieme, H.R. (2003) Mathematics in Population Biology. Princeton University Press.117

A mathematical analysis in discrete transition system (Theory of Biomathematics and its Applications IV)

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A mathematical analysis in discrete transition system (Theory of Biomathematics and its Applications IV)

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