BULLETIN O1~ MATIqEIYIATICAL BIOLOGY
VOLU~E 39, 1977
ALMOST CI~ITICAL ECOSYSTEMS
[] HEDLEY C. MORRIS
Department of Applied Mathematics, Trinity College, Dublin, Ireland
We introduce the concept of an almost critical system as one possessing certain critical stability properties when some of the small system parameters are placed equal to zero. We show how the location and stability properties of t h e critical points of almost critical systems can be deduced from those of the underlying critical system, providing the full mathematical criteria required.
i. Introduction. Often when we are trying to model a biological system with sets of differential equations our basic model tells us to put certain coefficients in our equations equal to zero. However, we feel t h a t in reality this cannot be correct and we include additional small terms which are outside of our original framework but which our conscience forces us to include. Are these really necessary ? Can we not obtain the majority of the information we require from our basic model equations ? As some of the system parameters are small it is legitimate to ask if, for certain purposes, we might be permitted to consider them to be actually zero. I f we are interested in the actual trajectories of the system, dynamic information, the answer is no, we m a y not. Small changes in the parameters of such a system can lead to discontinuous changes in the trajectory pattern. This is known as structural instability. Let us suppose, however, t h a t out interest is restricted, as it often is experimentally, to the consideration of steady-state configurations. Can we then, by studying the simpler system with the small term absent, learn anything about the biologically more interesting case when it is present ? In this paper we shall be primarily concerned with the case when a structural change does take place in the phase space trajectory patterns. Such a structural change will, in the cases we will 109
110
H E D L E Y C. MORRIS
look at, be related to the fact t h a t the resulting system is much easier to analyse and, as a result, it will, in practice, be the most common case. I f no simplification were to result there would be little point in making such a change. As a first illustration of these ideas we will in the next section analyse a model set of equations in two cases when a term is or is not present. When the term in question is absent the system has a degenerate form of equilibrium state in which there is an entire line of equilibrium points parallel to the axis of one of the system variables. This type of situation in which there are not discrete, isolated steady states, is known as critical stability. When the small term is present, the degeneracy is broken and one point on the line is singled out and moved to a point near to the critical line to define a unique steady state. 2. Model Equations.
Let us consider the model set of equations = p ( A n - B h - C),
(2.1)
= - p ( D n - E) + F - Gn,
(2.2)
= Hp-
(2.3)
8h2,
as typical of the sort of equations arrived at in modelling ecological systems. In fact, equations of exactly this structure form a model of J. Steel (1958; Riley, 1963) for the grazing of zooplankton h on phytoplankton p in a simplified two-layer model of the euphoric zone in the North Sea. This model considers the concentration of phasphate n as the key element controlling the photosynthetic rate of the phytoplankton. Particular details of such a model do not concern us for the moment but only the nature of the stable states of the system if the parameter 8 is small. We are going to consider the two cases s = 0 and 0 < 8 ~ 1 and compare our findings. How much will the ~ = 0 case be able to tell us about the second case when 0 < s ~ 1 ? (i) ~ = 0, the critical case. The equations for the equilibrium values of p, n and h become p ( A n - B h - C) = O,
(2.4)
p(Dn-E)
(2.5)
= F - Gn,
Tip = O,
(2.6)
which are easily solved to give =
0,
~ = F/G,
= C',
C'8 ( - oo, oo),
a whole line of critical points (0, F/G, C') parallel to the h-axis. Which of the points on this line are stable equilibrium points ? To determine this we linearise
ALMOST C R I T I C A L ECOSYSTEMS
111
about an a r b i t r a r y point (0, F / G , X). N e w coordinates are defined b y
~'
= p,
(2.~)
n" = n - F / G ,
(2.8)
h' = h - Z.
(2.9)
In terms of these equations (2.1)-(2.3) become, w i t h ~ = 0, 15" = - [C + B Z - A E / G ] p " + ( A p ' n ' - B p ' h ' ) ,
(2.10)
d" = - [ D F / G - E ] p ' - an" - D p ' n ' ,
(2.11)
]~" = H p ' .
(2.1.2)
Thus the linearisation equations which d e t e r m i n e t h e stability are given b y 13' n"
- (C + B X - A F / G ) =
-(DF/G-E)
h'
0
0
-G
0
o
o
H
p .
(2.13)
The secular e q u a t i o n for this m a t r i x is - 2(2 + G)(2 + C + B X - A F / G ) = O,
the zero root being an essential feature of critical stability. I f we ignore for the m o m e n t the p r o b l e m of the zero root we can see t h a t the R o u t h - H u r w i t z conditions (~Villems, 1970) reduce to (i) (ii)
G > 0, 21 = ( A F / G - B z - C )
When Z = ( ] / B G ) ( A F - G C )
< O,
Z >
(1/BG)(AF-GC).
(2.14)
we have a f u r t h e r degeneracy to two nullroots.
We shall see in a m o m e n t t h a t we do not need to consider the null root. (ii) 0 < 8 ~ l. The equations for the equilibrium values of p, n, h become= A~-B~-C
= O,
~ ( D ~ - E ) = ( F - G~),
(2.~5) (2.16) (~.17)
Solving these equations we find t h a t
(2.1~) n = (B/A)h+C/A,
and ~ is a root of the cubic e q u a t i o n
(2.19)
112
tIEDLEY
C.
MOlgiRIS
/DC-EA\-2
/HG\-
( G )
:o.
For small e the roots are given b y
f A F - GC~
A f A F - GC'~2/FD-EG'~
together with a complex conjugate pair C(s) = 8-~(-HGID)+ 0(s~).
(2.22)
As ~ -+ 0 +, hi(e) --~ ( ( A F - GC)/GB) the lower limit of the stable continuum. To investigate the stability of the equilibrium point
we must compute the Jacobian of the system equations at this point. This is given by
0
~A
-B~
-(DP~-E)
-(G+D~)
0
0
- 2ahl
H
(2.2 3
The secular equation is given by ;t8 + 22(G + D/5 + 2~h) + 2(28h(G + Dp) + (D~ - E)pA + BH~) + [2~h(D~- E)i)A + HB~(G + D/5)] = 0.
(2.24)
Let us retain only terms of order ~, then the equation becomes
+ eBG],2 = O,
(2.25)
where )7 = [(AF- GC)/GB]. The l~outh-Hurwitz conditions to order a are given by (i) (ii) (iii)
G+ e~(2+ D ~)>
O,
(2.26)
eBG],~ > O,
(2.27)
~\ -G-U ]A+2G > O.
(2.28)
A L M O S T C R I T I C A L ECOSYSTEMS
113
The condition G > 0 is clearly sufficient, especially if ~ > 0, to deal with (i). However, the other two represent an extension to the stability conditions on the critical system. Clearly the operations we have carried out here are simply representatives of a general procedure. I n fact the general equations show us more clearly w h a t is happening and show us t h a t provided the critical system underlying a real system is stable apart from a single null root and the additional terms are sufficiently small, there are only two additional conditions analogous to (ii) a n d (iii) which are required to preserve stability. L e t us find them. 3. The General Equations.
Suppose t h a t our ecosystem equations are given b y i = f(x, 8),
(3.1)
where x = @1 . . . . xn), ~ = (~1, • • . , %) are the system variables and the small coefficients of the system. The equations for the s t e a d y states are f(x, s) = O,
(3.2)
x = X(r).
(3.3)
which yield We suppose t h a t x(a) = 0(1) as ]]e]] --~0. L e t us concentrate our a t t e n t i o n on analytic solutions to this equation and suppose t h a t
x(~) = x ° + ~./~+0(11~fi2),
(3.4)
where f(x0, 0) = 0 a n d the dot p r o d u c t is in a-space. Substituting this into (3.2) the condition found for /~(i = 1, . . . , p) is (3.5)
\ 0 e ~ / x = x o ,~=0 ' where J t~¢°
\ ~Xj /x= ,,o ,a=o ,
(3.6)
the Jacobian of the critical system at a critical point. Hence j 0 is non-invertible° I t is this which fixes which point on the critical line is selected as lie11-30. j0 is a function of the p a r a m e t e r which defines the line, Z in the case we have been looking at. So also is (Dfk/aet),,=xO,a=o. There will be, because of the zero eigenvalue, necessary conditions t h a t (3.5) have a solution and this will provide an eigenvalue equation for Z. In the case considered eq. (3.5) becomes,
- (DF/G-
H H
E)
- G
0
0
0 _
fl~ _ f13
=
l
0 0 Z2
(3.7)
114
I - I E D L E Y C. M O R R I S
This only has a solution when X>0
if
(C+Bz-A.F/G)=
(3.8)
0
W e should note a t this point t h a t in order to determine 173 we m u s t go t o the n e x t highest order in e where a similar condition will fix it also. T o d e t e r m i n e t h e stability properties of these equilibrium points of the general s y s t e m we m u s t compute t h e J a c o b i a n of the s y s t e m at x = x(e). This is given b y =
(x0 +
=
+o(1i
a e t 0 x j J x=xO,~=o
j o j + ei L ~x~oaxJ
J~j(x(~)) ~ J ~o s + e.B~j+0(ll~II2).
ii2),
(3.9)
(3.10)
W e must now consider the secular e q u a t i o n for this matrix. Suppose t h a t it is given by, •~ n + p n - l ~ n - l + ... + P : ~ + P 0 = 0. (3.11) Now when e = 0, in the critical cases we are interested in, b o t h P l a n d P0 become zero b u t in general P n - 1 • • • p 2 t a k e on the non-zero values/9o_1, . . . ,p0. The H u r w i t z matrices (Willems, 1970) constructed f r o m the {p~}, D ° ( j = n - 2, . . . , i) are assumed positive and we suppose t h a t the e-corrections to t h e P n0- 1 . . . . . P20 will not affect t h e positivity of t h e D j ( j -= n - 2 . . . . , 1). This is certainly the case for sufficiently small ]]e[]. However, as P l a n d P0 are of first-order in ~ we m u s t obtain non trivial additional conditions from the two conditions D n - 1 > O,
D n > O.
(3.12)
T h u s we need the 0( lie ]]) corrections to P0 and P l 100 = (-- l) n det (JO+ e . B ) = ( - 1)n(det j o + e . t r B adj J°+0([]e]12)).t As det j o = 0 we h a v e Po = ( - 1)ns"tr [B adj JO]+0(lleI[2),
(3.13)
p l = ( - 1)n+l t r [adj ( j o + ~ . B ) ] = ( - 1)n+l[tr (~dj JO) + e. [tr JO t r B - t r (jOB)] + 0( lie ll2)]. I n our critical cases, t r (adj j o ) = 0 in general, and so p 1 = ( - 1)n+l e. [tr j o t r B - t r JOB] + 0( [[e [[2). ¢ W h e r e adj A is t h e a d j u g a t e m a t r i x of t h e m a t r i x A. T h e t r a n s p o s e of t h e m a t r i x of cofactors of A.
ALMOST
CRITICAL
ECOSYSTEMS
115
If we now retain only terms of order ]Iell in the last two I~outh-Hurwitz conditions (3.12) we have for n > 3 the two conditions (i) Ioo = (-- 1)n~ "tr (B.adj j0) > 0, (ii)
(PiP2°-P0P3)Dn-a° o > O.
Thus if se > 0 Y i we can say that the critical system is stable against such small perturbations if (i)
( - 1) n t r [B e adj j0] > 0, V i,
(ii)
( - 1)n+l{p°[tr J0 tr B ~ - t r (JOB)] + p o t r (B e adj j0)} > 0, V i.
4. Conclusions. We have considered systems which are 'almost critical' in the sense that if certain small coeffÉcients in the system equations are placed equal to zero the resulting system has critical stability properties. By means of an initial dydactic model example and then in general we have tried to show how such systems m a y be investigated by examining the simpler critical system which underlies them. Our aim has not been one of rigonr but of presenting an idea of sufficient generality to be useful. If, when we make inclusions 'by hand' into a well-defined set of model equations, we really do believe that they represent a small effect, why not remain with the original simpler model equations ? What we hope we have shown is the following:
(i) That the equilibrium states of the system, including the small terms, are within a distance of order equal to the magnitude of those small terms, from those of the critical system. The problem of which point of a critical line is selected by such perturbations has also been solved. (ii) That the stability of the almost critical system is largely controlled by the stability of the underlying critical system. The two extra conditions required to test full stability of the se]ected points from the critical lines have been constructed in general. The primary advantage of such a 'critical analysis' is that frequently the critical system underlying an almost critical system is capable of complete stability analysis without the aid of numerical methods and with none of the coefficients in the system having to be artificially placed equal to others to aid the computing problems involved. Often critical lines are completely independent of any of the system parameters. I should like to thank Professor T. D. Spearman for his interest and encouragement throughout the writing of this paper.
116
t:IEDLEY C. MORRIS LITERATURE
Riley, C. A. 1963. The Sea, ed. M. N. Hill. Interscienee Publishers. Steel, J. H. 1958. "P]~nt Production in the Northern North Sea." Scottish Home Dept. of Marine Research 7, 1-36. Willems, J. L. 1970. Stability Theory of Dynamical Systems. Nelson. RECEIVED 5-19-73 REVISED 5-9-76
VOLU~E 39, 1977
ALMOST CI~ITICAL ECOSYSTEMS
[] HEDLEY C. MORRIS
Department of Applied Mathematics, Trinity College, Dublin, Ireland
We introduce the concept of an almost critical system as one possessing certain critical stability properties when some of the small system parameters are placed equal to zero. We show how the location and stability properties of t h e critical points of almost critical systems can be deduced from those of the underlying critical system, providing the full mathematical criteria required.
i. Introduction. Often when we are trying to model a biological system with sets of differential equations our basic model tells us to put certain coefficients in our equations equal to zero. However, we feel t h a t in reality this cannot be correct and we include additional small terms which are outside of our original framework but which our conscience forces us to include. Are these really necessary ? Can we not obtain the majority of the information we require from our basic model equations ? As some of the system parameters are small it is legitimate to ask if, for certain purposes, we might be permitted to consider them to be actually zero. I f we are interested in the actual trajectories of the system, dynamic information, the answer is no, we m a y not. Small changes in the parameters of such a system can lead to discontinuous changes in the trajectory pattern. This is known as structural instability. Let us suppose, however, t h a t out interest is restricted, as it often is experimentally, to the consideration of steady-state configurations. Can we then, by studying the simpler system with the small term absent, learn anything about the biologically more interesting case when it is present ? In this paper we shall be primarily concerned with the case when a structural change does take place in the phase space trajectory patterns. Such a structural change will, in the cases we will 109
110
H E D L E Y C. MORRIS
look at, be related to the fact t h a t the resulting system is much easier to analyse and, as a result, it will, in practice, be the most common case. I f no simplification were to result there would be little point in making such a change. As a first illustration of these ideas we will in the next section analyse a model set of equations in two cases when a term is or is not present. When the term in question is absent the system has a degenerate form of equilibrium state in which there is an entire line of equilibrium points parallel to the axis of one of the system variables. This type of situation in which there are not discrete, isolated steady states, is known as critical stability. When the small term is present, the degeneracy is broken and one point on the line is singled out and moved to a point near to the critical line to define a unique steady state. 2. Model Equations.
Let us consider the model set of equations = p ( A n - B h - C),
(2.1)
= - p ( D n - E) + F - Gn,
(2.2)
= Hp-
(2.3)
8h2,
as typical of the sort of equations arrived at in modelling ecological systems. In fact, equations of exactly this structure form a model of J. Steel (1958; Riley, 1963) for the grazing of zooplankton h on phytoplankton p in a simplified two-layer model of the euphoric zone in the North Sea. This model considers the concentration of phasphate n as the key element controlling the photosynthetic rate of the phytoplankton. Particular details of such a model do not concern us for the moment but only the nature of the stable states of the system if the parameter 8 is small. We are going to consider the two cases s = 0 and 0 < 8 ~ 1 and compare our findings. How much will the ~ = 0 case be able to tell us about the second case when 0 < s ~ 1 ? (i) ~ = 0, the critical case. The equations for the equilibrium values of p, n and h become p ( A n - B h - C) = O,
(2.4)
p(Dn-E)
(2.5)
= F - Gn,
Tip = O,
(2.6)
which are easily solved to give =
0,
~ = F/G,
= C',
C'8 ( - oo, oo),
a whole line of critical points (0, F/G, C') parallel to the h-axis. Which of the points on this line are stable equilibrium points ? To determine this we linearise
ALMOST C R I T I C A L ECOSYSTEMS
111
about an a r b i t r a r y point (0, F / G , X). N e w coordinates are defined b y
~'
= p,
(2.~)
n" = n - F / G ,
(2.8)
h' = h - Z.
(2.9)
In terms of these equations (2.1)-(2.3) become, w i t h ~ = 0, 15" = - [C + B Z - A E / G ] p " + ( A p ' n ' - B p ' h ' ) ,
(2.10)
d" = - [ D F / G - E ] p ' - an" - D p ' n ' ,
(2.11)
]~" = H p ' .
(2.1.2)
Thus the linearisation equations which d e t e r m i n e t h e stability are given b y 13' n"
- (C + B X - A F / G ) =
-(DF/G-E)
h'
0
0
-G
0
o
o
H
p .
(2.13)
The secular e q u a t i o n for this m a t r i x is - 2(2 + G)(2 + C + B X - A F / G ) = O,
the zero root being an essential feature of critical stability. I f we ignore for the m o m e n t the p r o b l e m of the zero root we can see t h a t the R o u t h - H u r w i t z conditions (~Villems, 1970) reduce to (i) (ii)
G > 0, 21 = ( A F / G - B z - C )
When Z = ( ] / B G ) ( A F - G C )
< O,
Z >
(1/BG)(AF-GC).
(2.14)
we have a f u r t h e r degeneracy to two nullroots.
We shall see in a m o m e n t t h a t we do not need to consider the null root. (ii) 0 < 8 ~ l. The equations for the equilibrium values of p, n, h become= A~-B~-C
= O,
~ ( D ~ - E ) = ( F - G~),
(2.~5) (2.16) (~.17)
Solving these equations we find t h a t
(2.1~) n = (B/A)h+C/A,
and ~ is a root of the cubic e q u a t i o n
(2.19)
112
tIEDLEY
C.
MOlgiRIS
/DC-EA\-2
/HG\-
( G )
:o.
For small e the roots are given b y
f A F - GC~
A f A F - GC'~2/FD-EG'~
together with a complex conjugate pair C(s) = 8-~(-HGID)+ 0(s~).
(2.22)
As ~ -+ 0 +, hi(e) --~ ( ( A F - GC)/GB) the lower limit of the stable continuum. To investigate the stability of the equilibrium point
we must compute the Jacobian of the system equations at this point. This is given by
0
~A
-B~
-(DP~-E)
-(G+D~)
0
0
- 2ahl
H
(2.2 3
The secular equation is given by ;t8 + 22(G + D/5 + 2~h) + 2(28h(G + Dp) + (D~ - E)pA + BH~) + [2~h(D~- E)i)A + HB~(G + D/5)] = 0.
(2.24)
Let us retain only terms of order ~, then the equation becomes
+ eBG],2 = O,
(2.25)
where )7 = [(AF- GC)/GB]. The l~outh-Hurwitz conditions to order a are given by (i) (ii) (iii)
G+ e~(2+ D ~)>
O,
(2.26)
eBG],~ > O,
(2.27)
~\ -G-U ]A+2G > O.
(2.28)
A L M O S T C R I T I C A L ECOSYSTEMS
113
The condition G > 0 is clearly sufficient, especially if ~ > 0, to deal with (i). However, the other two represent an extension to the stability conditions on the critical system. Clearly the operations we have carried out here are simply representatives of a general procedure. I n fact the general equations show us more clearly w h a t is happening and show us t h a t provided the critical system underlying a real system is stable apart from a single null root and the additional terms are sufficiently small, there are only two additional conditions analogous to (ii) a n d (iii) which are required to preserve stability. L e t us find them. 3. The General Equations.
Suppose t h a t our ecosystem equations are given b y i = f(x, 8),
(3.1)
where x = @1 . . . . xn), ~ = (~1, • • . , %) are the system variables and the small coefficients of the system. The equations for the s t e a d y states are f(x, s) = O,
(3.2)
x = X(r).
(3.3)
which yield We suppose t h a t x(a) = 0(1) as ]]e]] --~0. L e t us concentrate our a t t e n t i o n on analytic solutions to this equation and suppose t h a t
x(~) = x ° + ~./~+0(11~fi2),
(3.4)
where f(x0, 0) = 0 a n d the dot p r o d u c t is in a-space. Substituting this into (3.2) the condition found for /~(i = 1, . . . , p) is (3.5)
\ 0 e ~ / x = x o ,~=0 ' where J t~¢°
\ ~Xj /x= ,,o ,a=o ,
(3.6)
the Jacobian of the critical system at a critical point. Hence j 0 is non-invertible° I t is this which fixes which point on the critical line is selected as lie11-30. j0 is a function of the p a r a m e t e r which defines the line, Z in the case we have been looking at. So also is (Dfk/aet),,=xO,a=o. There will be, because of the zero eigenvalue, necessary conditions t h a t (3.5) have a solution and this will provide an eigenvalue equation for Z. In the case considered eq. (3.5) becomes,
- (DF/G-
H H
E)
- G
0
0
0 _
fl~ _ f13
=
l
0 0 Z2
(3.7)
114
I - I E D L E Y C. M O R R I S
This only has a solution when X>0
if
(C+Bz-A.F/G)=
(3.8)
0
W e should note a t this point t h a t in order to determine 173 we m u s t go t o the n e x t highest order in e where a similar condition will fix it also. T o d e t e r m i n e t h e stability properties of these equilibrium points of the general s y s t e m we m u s t compute t h e J a c o b i a n of the s y s t e m at x = x(e). This is given b y =
(x0 +
=
+o(1i
a e t 0 x j J x=xO,~=o
j o j + ei L ~x~oaxJ
J~j(x(~)) ~ J ~o s + e.B~j+0(ll~II2).
ii2),
(3.9)
(3.10)
W e must now consider the secular e q u a t i o n for this matrix. Suppose t h a t it is given by, •~ n + p n - l ~ n - l + ... + P : ~ + P 0 = 0. (3.11) Now when e = 0, in the critical cases we are interested in, b o t h P l a n d P0 become zero b u t in general P n - 1 • • • p 2 t a k e on the non-zero values/9o_1, . . . ,p0. The H u r w i t z matrices (Willems, 1970) constructed f r o m the {p~}, D ° ( j = n - 2, . . . , i) are assumed positive and we suppose t h a t the e-corrections to t h e P n0- 1 . . . . . P20 will not affect t h e positivity of t h e D j ( j -= n - 2 . . . . , 1). This is certainly the case for sufficiently small ]]e[]. However, as P l a n d P0 are of first-order in ~ we m u s t obtain non trivial additional conditions from the two conditions D n - 1 > O,
D n > O.
(3.12)
T h u s we need the 0( lie ]]) corrections to P0 and P l 100 = (-- l) n det (JO+ e . B ) = ( - 1)n(det j o + e . t r B adj J°+0([]e]12)).t As det j o = 0 we h a v e Po = ( - 1)ns"tr [B adj JO]+0(lleI[2),
(3.13)
p l = ( - 1)n+l t r [adj ( j o + ~ . B ) ] = ( - 1)n+l[tr (~dj JO) + e. [tr JO t r B - t r (jOB)] + 0( lie ll2)]. I n our critical cases, t r (adj j o ) = 0 in general, and so p 1 = ( - 1)n+l e. [tr j o t r B - t r JOB] + 0( [[e [[2). ¢ W h e r e adj A is t h e a d j u g a t e m a t r i x of t h e m a t r i x A. T h e t r a n s p o s e of t h e m a t r i x of cofactors of A.
ALMOST
CRITICAL
ECOSYSTEMS
115
If we now retain only terms of order ]Iell in the last two I~outh-Hurwitz conditions (3.12) we have for n > 3 the two conditions (i) Ioo = (-- 1)n~ "tr (B.adj j0) > 0, (ii)
(PiP2°-P0P3)Dn-a° o > O.
Thus if se > 0 Y i we can say that the critical system is stable against such small perturbations if (i)
( - 1) n t r [B e adj j0] > 0, V i,
(ii)
( - 1)n+l{p°[tr J0 tr B ~ - t r (JOB)] + p o t r (B e adj j0)} > 0, V i.
4. Conclusions. We have considered systems which are 'almost critical' in the sense that if certain small coeffÉcients in the system equations are placed equal to zero the resulting system has critical stability properties. By means of an initial dydactic model example and then in general we have tried to show how such systems m a y be investigated by examining the simpler critical system which underlies them. Our aim has not been one of rigonr but of presenting an idea of sufficient generality to be useful. If, when we make inclusions 'by hand' into a well-defined set of model equations, we really do believe that they represent a small effect, why not remain with the original simpler model equations ? What we hope we have shown is the following:
(i) That the equilibrium states of the system, including the small terms, are within a distance of order equal to the magnitude of those small terms, from those of the critical system. The problem of which point of a critical line is selected by such perturbations has also been solved. (ii) That the stability of the almost critical system is largely controlled by the stability of the underlying critical system. The two extra conditions required to test full stability of the se]ected points from the critical lines have been constructed in general. The primary advantage of such a 'critical analysis' is that frequently the critical system underlying an almost critical system is capable of complete stability analysis without the aid of numerical methods and with none of the coefficients in the system having to be artificially placed equal to others to aid the computing problems involved. Often critical lines are completely independent of any of the system parameters. I should like to thank Professor T. D. Spearman for his interest and encouragement throughout the writing of this paper.
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t:IEDLEY C. MORRIS LITERATURE
Riley, C. A. 1963. The Sea, ed. M. N. Hill. Interscienee Publishers. Steel, J. H. 1958. "P]~nt Production in the Northern North Sea." Scottish Home Dept. of Marine Research 7, 1-36. Willems, J. L. 1970. Stability Theory of Dynamical Systems. Nelson. RECEIVED 5-19-73 REVISED 5-9-76
