J. theor. Biol. (1975) 50, 3-3
Global Asymptotic Stability Criteria for Models of Density-dependent Population Growth D. L. DE ANGELS Imtitufe for Soil Science and Forest Fertilization, Goettingen University, Goettingen, West Germany (Received 25 March 1974) A single-population, fertility or mortality of the population or determine conditions
multiple-age-class model is formulated in which the rates are dependent on the total biomass density a related function. The circle criterion is applied to for which the model is globally asymptotically
stable about an equilibrium age distribution.
1. IntrodM!tion Density-dependent effects are generally recognized to play an important role in the regulation of population size. Density dependence in population growth has been defined by Murdoch (1970) to occur “when the rate of increase of the population tends to be a decreasing function of density . . .Y’, and he adds that “density dependence is necessary for stability, but unless it is powerful enough it may not be sufficient” (Murdoch, 1970). In cases where it is not sufficient, population densities may reach extremely high levels and subsequently crash because of mass starvation. Stability is the tendency of a system to resist such catastrophes. This can be expressed more precisely by assuming the dynamics of an ecological system to be governed by a set of (in general, non-linear) differential equations, with the dependent variables representing population densities. The set of equations is stable if the variables tend to return to particular equilibrium values following a perturbation. If a single species is considered, it is said to have a stable age distribution if the variables representing the population densities of various age classes always return to equilibrium values following a perturbation. Two types of stability may be distinguished. Local stability is stability with respect to infinitesimally small perturbations about the equilibrium point, whereas a system exhibits global asymptotic stability tihen its variables return to the equilibrium point following a perturbation of arbitrary magni35
tude. Many populations are subject to sudden perturbations of one or more orders of magnitude. In such cases local stability may have little meaning and it is logical to inquire whether mathematical models for such species have global asymptotic stability. Published computer simulations of single-species population models have included density-dependent functions to regulate the population growth (King & Paulik, 1967; Pennycuick, Compton & Beckingham, 1968). Though useful for studying particular models, computer simulations do not usually lead to broad generalizations concerning such regulation, for which more formal mathematical methods are necessary. Some developments in the theory of non-linear differential equations during the last decade or so provide global stability criteria appropriate to certain equations of interest in population dynamics. This paper considers some implications of a particular stability criterion, the circle criterion, to a single-species model in which the population is divided among age classes having different biomasses, fertility rates and survival rates. Fertility and mortality rates are assumed dependent on the total biomass density of the species. We show that this criterion is useful in establishing parameter value conditions which are sufficient for stability. The next section will introduce the model studied. The section following it briefly describes the circle criterion, while in the final section this criterion is applied to the population model. 2. Single Species Population Model
Consider a species divided into n age groups. It is often convenient to divide the species into reproductive or other physiological stages, characterized by different fertility and survival rates. If the ages of the organisms are actually distributed in a more or less continuous fashion, one can, to good approximation, use differential equations instead of Leslie matrices (Williamson, 1972) to simulate the stransfer of organisms from one discrete age class to the next. We will use the differential equation model here, keeping in mind that it is not appropriate to all species. If both fertility and mortality rates of the species are assumed to be dependent on biomass density, a mathematical model for its behaviour can be written as follows: . Xl = b(A!f)(f, x1 + . . . +.h~ -G-1 + dlW)h . x2 = ~lrl~l-(r,+Mwh . . . . x, = z,-lr,-lx,-l-(I;,+d,(M))x, (1)
where, for i = 1, n: xi, i’r = population number density of the ith age class, and the corresponding time derivative. ri = rate at which organisms leave ith age class in absence of density-dependenteffects. f, = fraction of those organisms leaving ith age class which enter (i+ l)st ageclass(i.e. survival coefficient). A= rate of reproduction of the ith ageclass. M = total biomass of species; M = c mixi, where m1 is the average biomass of an organism in the ith age class. 4w), mf) = arbitrary positive functions of M, which represent the dependence of the mortaility and birth rates, respectively, on iU. In a more general senseone can interpret M as a stressfactor which is related to the number density of individuals, but which may or may not be proportional to biomass density. The coefficient mi is then a measure of the contribution an organism of the ith age classmakes to the total stress on population growth. For purposes of further analysis, it is convenient to cast (1) in a form such that the equilibrium point of interest (i.e. the non-trivial equilibrium point, x10, x20, * * *, xno # 0) is transformed to the origin of the new co-ordinate system.This is accomplishedby substituting Xl
x2 = x2o+y2 . .
&I = x*0+ Yw Inserting (2) into (1) and employing the equilibrium conditions, we obtain the following set of equations: j* = M’.b’(M’)(f,.(x,,+y,)+. . . +f, *(x.0 + Y”N + WOXfl Yl + * 0 * +f,YJ - h + 4WoNY 1 - M’*(x,,+Y1)4(~‘) 32 = h ri Yi - (r2 + dz(Mo))Yz - M’ . (X20 + Y,&(M’)
E (b(M, + M’) - b(M,))/W
+ M’) - d(M,))/M
The equations written in this form have meaning only when b(M,,+M’) and d,(M, + M’) are continuous functions of M’ near M’ = 0. 3. A Stability
Criterion for Non-linear
It is possible to derive an important criterion for the global asymptotic stability of systems of non-linear differential equations. The derivation, based on the construction of a Liapunov function, is not discussed here, but is given by Willems (1973). Let D = d/dt and p(D) E D”+p,-,lv+. . . +po q(D) E qn-JY-‘+. W)
. . +qo
where qr and pi are constants. If a non-linear equation has the form: P(@Y + k . CIP)Y = 0 (6) is an arbitrary function of its own argument, where k(y, Dy, . . ., D”-‘y) the circle criterion states that the equation is globally asymptotically stable if the following two conditions are satisfied: k, < k < k, (7) (0 where
for all values of 0. (ii) The values of s for which p(s)+k.q(s)
must have negative real parts for all values of k in in the range k, < k < kz. It will be shown in the next section that in certain cases equation (3) can be put into the form of equation (6). Then the above conditions can be applied to test the global asymptotic stability of the population model. 4. Applications
The applications of the circle criterion can be exhibited by some concrete examples. In the interest of avoiding cumbersome mathematics, the examples will be restricted to cases where only three age groups are considered and where the age group transition rates, rf, the survival percentages, Zi, are assumed to have the same values, r and 1 respectively, for each age group.
The youngest age group will be assumed sexually immature in all examples, so that fi = 0. These assumptions simplify calculations considerably, but the criterion can be used just as easily when they are not made. The circle criterion is conveniently applied only when just one of the non-linear functions b(M), d,(M), or d,(M) are present in the model at one time. Otherwise the mathematics becomes formidable. Although this consideration limits the usefulness of the criterion, the study of simple examples can often provide insight into more complex ones. The following three cases illustrate this. case 1: d&w’) = d*(W) = d,(M’) = 0. In this case the population is regulated by the density-dependence of its fertility rate, but none of the age groups exhibit density-dependent mortality rates. With the above simplifications, the set of equations (3) can be reduced to a single third-order equation, I3
+ 3v3 + (3rZ - kf2
+ (r3 -vi
where PO E ml r2 + m, lr2 + m3 12r2 PI z 2m,r+lrmz f12 z ml.
Equation (10) has the same form as equation (a), with (lla) P(s) = s3 +3rs2+(3r2 - Zrfi b(M,))s + r2(r - j(f2 + u3)b(M,)) 4(s) = -B232-Bim30 UW k = (f2. (X20 + Y2) +f3. (x30 +y3))b’(M’). WC) For equation (10) to be globally asymptotically stable, it is sufficient that equation (7) and (8) are satisfied and that equation (9) is true only when s has no positive real parts. When (11) is substituted into equation (9) this last condition is satisfied when 3r2-rZf2b(Mo)
r - U2 -t Y3M40) = 0 (124 and when inequalities (17a) and (17b) (below) hold. In this case equation (9) reduces to a third-order polynomial in s the real parts of the roots of which are always negative. This can be shown from the Routh-Hurwitx criterion (Kom & Kom, 1968). The condition (12c) is satisfied since one of the
on equation (3) for case 1 is r - Z(f2 + If,)b(M,) = 0. (13) From equation (13) it is a trivial matter to show that equation (12b) is also satisfied. Since fi .(x2,, +y,) +f3. (xSO+y,) > 0, condition (12a) requires that b’(M’) = (b(M, + M’) - b(M,))/M’ < 0. (14) This is true when b(M) is a continuously decreasing function of the total species biomass, M. It can also hold true for cases in which b(M) has more complex behaviour, but b(M) certainly must be decreasing on the average. The above results have disclosed that we should require b’(M’) < 0 for stability. It follows then from condition (7) and equation (11~) that kr < k, < 0. It is desirable to choose the range of permitted values of k to be as broad as possible. Therefore we choose k, = 0 and k, + - 00, in which case condition (8) reduces to Re (G(io)) < 0. (15) Substitution of condition (1 la) and (1 lb) into (15) results in the following condition for global asymptotic stability: c,w2+c404
where C2 = B1(3r2 - lrf2 b(M,)) - 3r/l, c4 E 3rfi2-j&.
From equation (16) it is obvious that stability is guaranteed if C,, C, > 0, or (3r-21f, b(Mo))ml > 3Z2rm, + lzf2 b(M,)m, (174 m, > Im,. tl7b) In view of condition (12~) it can be shown that 3r -2lf,b(M,-,) is always positive. The two inequalities (17a) and (17b) therefore state that the population stress coefficient m, should not be less than m2 and m3 multiplied by certain constants which are functions of other system parameters. The greater the contribution of the individual organisms of the youngest age group to the overall stress on the birth rate, the more stable the system will be. It is also evident from the inequalities that small survival fractions, Z, and fertility rates, f2, promote stability. The implications on b(M), ml, m2 and m3 noted in the above analysis seem intuitively reasonable. But the particular inequalities (17a) and (17b) are certainly not obvious, which in part justiRes this type of analysis. It is emphasized that these conditions are sufficient, but not necessary, for global asymptotic stability. Many parameter values which violate these
cmditions can still result in a stable system. Thorough analysis of equation (16) would lead to a broader range of permissible parameter values. case 2: b’(M’) = 1, d;(M) = #j(M) = 0. This case corresponds to the situation where only members of the youngest age group exhibit a mortality rate which is sensitive to population density effects. No density dependence of the fertility rate is assumed. Following the procedure used for Case 1, we find that sufficient conditions for global asymptotic stability are d;(M’) > 0, (18) i.e. d,(M) should be an increasing function of it4, and c;02+c;04> 0, (19) where C; = B;r(3r+2dl(M,)-If,)-B~(3r+dl(M,)) Ci 5 (3r+d,(Mo))ml-j3; &)E r2(ml +m2 I+m3Z2) B; = r(2mi+m21). For inequality (19) to hold, it is sufficient that CL, Ci > 0; that is, Mr+4Wd)-W2h > r2Z(lfi-dltM,))m2+Z2(3r+dl(Mo))m3 (204 (r+f&(M,))m, > Zrm2. t2W The two inequalities (20a) and (2Ob) imply certain relationships among the stress coefficients, mt. Inequality (2Ob) states that m, should be larger than some constant (which is less than unity, since I < 1) multiplied by m,. It can be shown from the equilibrium condition that the coefficient of m, in (20a) is positive. The coefficient of m2 may be either positive or negative. Hence, depending on the sign of this coe5cient, su5cient conditions for stability require either (a) m1 be not less than the sum of m2 and m, multiplied by coe5cients depending on other system parameters, or (b) a certain linear combination of m, and m, (both temx positive) cannot be less than m3. Clearly, stability is most favoured when mi is very large relative to m3 and m,; that is, when the mortality rate of the youngest age group is most strongly influenced by its own population density. Again it is noted that stability is favoured when I andf2 are small. case3: b’(A4’) = 1, d;(W) = d;(W) = 0. In this case only the oldest age group experiences a mortality rate which depends on the total population stress, and again no regulation of the birth
rate is exercised. Following the procedure used in the first two cases, we find the sufficient conditions for global asymptotic stability to be dj(M’) c;u2+c;w4>
> 0 0
where cl; = B; r(3r+2d,(M0)-If,)-PA(3r+d,(M,)) Cl = (3r + d,(M,))m3
/Xi = rj~m,+lrf,m,+r(r-If,)m,
8’; -f3m,f2rm3. Again, (20) is satisfied if we assume C;, Ce > 0, which implies the following inequalities :
The inequality (23b) implies that m3 should be larger than some constant multiplied by m,. The coefficient of m, in (23a) can be either positive or negative. Therefore, this inequality implies either that m3 is larger than some linear combination of m, and m2, or that a linear combination of m, and m, is larger than m,. There is a strong similarity between these results and the results of case 2. Here the system tends to be more stable when the oldest age group exerts the greatest stress on its own mortality rate, while in case 2 stability was enhanced when the youngest age group contributed the most stress on its own mortality rate. Stability is more likely when f3 is small, but it is difficult to draw any generalizations concerning I and ft in this case. More complex cases than those above can be investigated using the circle criterion if the equations of the system can be reduced to a single equation of the form of equation (6). In general, this is extremely complicated when the non-linear functions, d,(M), occur in other than the youngest or oldest age classes, although the mathematics may be tractable if simple enough functions are chosen for d,(M). 5. Conclusion
The principle results of this paper are embodied in the inequalities (14), (16)-(23) which state conditions sufficient for the global asymptotic stability of certain singIe-species models whose fertility 6r mortality rates are influenced by population density. Two general rules of thumb emerge from these formulae:
(1) If the fertility rate is density-dependent, the system is better regulated toward stability if the youngest age group exerts, per individual organism, the strongest stress on the birth rate. (2) If the population density is regulated by means of a density-dependent mortality rate for the youngest (oldest) age class, regulation towards stability is most effective when the youngest (oldest) age class has, per unit individual organism, the greatest influence on its own mortaility. The second rule of thumb strongly suggests a more general conclusion, namely, that a population is most stable when the density-dependent regulation acting on a particular age class depends most strongly on the density of that particular age class. This can be shown to be true, at least for local stability, by examination of the matrix of perturbed equations. The more strongly the density-dependent regulation of the ith age class itself, the more negative the diagonal elements of the matrix will be as opposed to the off-diagonal elements; this promotes stability. The author was supported by the German Research Association (DFG), under contract No. El l/37.
KINO, C. E. &
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