Rev. Acad. Canar. Cienc., XVIII (Núms. 1-2), 33-45 (2006) (publicado en agosto de 2007)
GENERALIZATION OF REPLICATOR DYNAMICS
FROM A 3-SIMPLEX TO A 2-SPHERE WITH A BOUNDARY
A. Gupta* & K. Bhattacharya
Department of Pure Mathematics
University of Calcutta
ABSTRACT. The present paper extends the idea of replicator dynamical
system from a 3-simplex to a 2-sphere with boundary. It
derives a necessary and also a suffi.cient condition of permanence of
such generalized replicator dynamical system on a compact part of
S2 with boundary.
Key words and phrases Permanence, Generalized Replicator equation,
Average Lyapunov function, Concentration simplex, Inhomogeneous
hypercycle.
Ü. lNTRODUCTION
In biochemistry or biology there is one class of molecules, which appeared
one day during evolution, for which selfreplication is obligatory.
These molecules, of course, are the polynucleotides, the nucleic acid or,
later in evolution the genes. These molecules interact with each other
(rather the dynamics or selfreplication between two or more molecules
happens) in a reaction vessels called evolution reactor. The evolution reactor
is a kind of flow reactor which consists of reaction vessel and allows
for temperature and pressure control. Its walls are impermeable to the
self replicative units (biological macromolecules like polynucleotides-e.g.
phage RNA- bacteria or in principle, also higher organisms). Energy rich
* Bengal College of Engineering and Technology, Durgapur-12, INDIA.
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material (food) is poured from the environment into the reactor. The
degradation products (waste) are removed steadily. In such a evolution
reactor, material support is so adjusted that the food concentration remains
constant in the reactor, the waste is drawn out by dilution flux.
n
Mathematically it means that L Xi = e, if xi are the concentrations.
i=l
So any differential equation (preferably replicator type of equation) in-volving
such x/s means that the state space is a n- simplex. This is
why n- simplex is also called a concentration simplex. It may be seen
n n n n
from the equality L x; = (L xi)2 - 2 L XiXj that even if L xi is not
i=l i=l i,j=l i=l
n n
constant, L XiXj may so adjust among themselves that L x; remains
i,j=l i=l
constant. So it may be more convenient to adjust the material support
in order to keep the sum of the squares of the concentrations constant.
This means mathematically to develop replicator differential equation
n
on such a surface on which L x; remains constant. This suggests pref-i=
l
erence in considering surface of a sphere in place of simplex as the state
space for replicator system of dynamics. In particular, to study permanence
criterion of such differential equation on the surface of a 2-
dimensional sphere with boundary deserves special attention.
There are two alternatives in the dynamics of a species on Rn viz.
either it extincts or it survives, in the sense that, it either hits or does
not hit the boundaries which are the positive axes of the coordinates.
The system is called permanent if no species becomes extinct. Permanence
can also be treated as the repelling property of the boundary;
more precisely it means that there exists a compact set K in the interior
of R+ such that every orbit starting at a point x in the interior
satisfies x(t) E K if t --+ +oo. In otherwords, whenever xi (O) > O
for i = 1, 2, ..... , n, there exist constants a, A, O < a < A such that
a < liminfxi(t) < limsupxi(t) < A for all i.
t->OO t ->OO
Sorne mathematicians [1,2,3] extended the idea of permanence and
extinction from Rn to a locally compact metric space. They obtained
a sufficient condition of permanence/uniform persistence of fiows on a
locally compact metric space or that on a complete metric space, the
boundary of the state space being invariant under the fiow. In this connection,
two natural queries may arise:
(1) Does there exist a nontrivial example of a locally compact metric
space/a complete metric space with"boundary anda corresponding fiow
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which keeps the boundary invariant? (2) Is the fiow a global fiow? In
fact , the condition of permanence becomes useless unless satisfactory
answer to query (1) can be given; also the stability analysis becomes
meaningless unless condition (2) is justified. In this context it may be
mentioned that sorne mathematicians have chosen the state space as
a n- simplex or a concentration simplex or a n- polygon [4,5,6] and
have found out such a form of system of differential equations (replicator
system) which keeps the boundary and faces of the n- simplex
( concentration simplex) or a n- polygon invariant. For such a choice,
the outcome is two fold. Firstly, the state space being compact, it
is ensured that the fiow is a global fiow which is a must for stability
analysis. Secondly, from the very construction of Replicator system, it
follows that the boundary of the state space remains invariant in such
cases. Thus a replicator system on a n-simplex is a concrete example
where both the aforesaid queries are met with satisfactorily. Again it
is not out of context to mention that such replicator system on n- simplex
5n [4] plays an important role in theoretical biology, e.g. ecology,
genetics, evolutionary game theory or chemical kinetics [6 ,7]. A more
general replicator system on n- concentration simplex 5~ [6] also plays
an important role for chemical and biological species. It is noted that
a simplex consists of plane surfaces only. N aturally a question may be
raised as to generalization of replicator system and its dynamics on a
curved surface. A convenient form of such surface is naturally that of a
2- sphere. With this motivation, the present paper develops the theory
of replicator system on another particular type of compact set, other
than simplex viz. a part of 52 with boundary. Moreover, it is not out
of context to mention that there have been later attempts also on permanence
criteria in general system but sorne interesting review works
might be available, but they are not directly related to our present work.
In this paper, at first we give sorne known definitions and results of
replicator systems. Then we give the idea of generalized replicator equation
on 2- sphere with boundary and obtain a necessary and sufficient
criterion of permanence of such generalized replicator system on 52 with
boundary.
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l. 80ME KNOWN DEFINITIONS AND RESULTS [4,5,6]
Definition 1.1 [4] Let a dynamical system on
n
Sn = {x = (x1, X2, .... .. . , Xn) E Rn: L Xi= 1, Xi~ O, Vi} be given by
i=l
n
where f = L xdi ( x). ( I) is called a replicator equation. It lea ves the
i=l
boundary and all faces of Sn invariant.
In general, the function Íi(x) is linear in Xi for i = 1, 2, .... , n.
Definition 1.2 [4] The dynamical system (J) on Sn is called permanent,
if bdSn is a repeller i.e. there exists M >O such that if xi(O) > O,
Vi, then xi(t) > M, for all sufficiently large t and Vi= 1, 2, ..... , n.
Definit ion 1.3 [4] Let x(t) be the orbit of (J) with x(O) = x . Let
P : Sn ---+ R be a differentiable function on Sn which satisfies
(1) P(x) =O Vx E bdSn
(2) P(x) > O Vx E intSn
and further
(3) P =P. iJ!(x)
where iJ! is continuous function on Sn. Such a function P is called an
average Lyapunov function on Sn.
Result 1.1 [4,5] The replicator equation (J) on Sn is permanent if
there exists a diff erentiable function P : Sn ---+ R such that the following
two conditions hold:
(i) For x E Sn, ~(~~ = iJ!(x)
(ii) For x E bdSn, ~ J;{ iJ!(x(t))dt > O for sorne T >O.
Remark 1.1 [4] If Sn is replaced by any compact set X and bdSn be
a closed invariant subset A of X, then also the Result 1.1 holds good.
This shows that the Result 1.1 is a general result which is applicable
to any compact set with boundary kept invariant under the flow of the
given system.
An alternative sufficient condition of permanence of replicator equation
is given as follows:
Result 1.2 [4] If P satisfies (1), (2) , (3) and if every orbit on the
boundary of Sn converges to a fixed point and if iJ! ( x) > O for all such
fixed points on the boundary, then (J) is permanent.
Remark 1.2 That the condition of result 1.1 ensures the permanence
may be explained as follows:
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In fact, this condition implies that \II > O on bdSn- So P(x) > O in
intSn. This means that P(x) is an increasing function. As P(x) denotes
the distance from the boundary, so the orbit will be repelled from the
boundary. In otherwords, the system becomes permanent.
Definition 1.4 [6] Let
n
s~ = {x = (x1,X2, .... ,xn) E Rn: ¿xi= c,xi::::: o far 1:::; i:::; n}. It is
i=l
called the concentration simplex.
The dynamics on S~ is given by the differential equations
(I I) Xi = xi[qi + ¿J=l kijXj - ~]
n n
where e > O, Qi and kij E R and </> = L.: xi(Qi + L kijXj)· The xi
i=l j=l
represents the concentration of the chemical or biological species i and
Qi E R corresponds to the selfreproduction or decay of the species i and
kijXj represents the effect of the species j on the reproduction of species
i which is of mass action type, catalytic kij > O and inhibiting kij < O.
(I I) is called a replicator system on S~; if it keeps the boundaries and
faces of s~ invariant.
Remark 1.3 The particular case of (I I) is the inhomogeneous
hypercycle given by
(I I I) xi = xi[qi + kixi-1 - <I>]
ki > O, i = 1, 2, .... , n. Here the species 1 acts catalytically on 2, 2 on
3, 3 on 4 and so on. This equation was studied in [6]. The following
result was obtained there: The system is cooperative if the selfreplication
terms Qi are sufficiently small i.e. it is only slight perturbations
of the homogeneous hypercycle (all Qi = O) for which the cooperation/
permanence was shown in [8]. On the other side (III) and more
general systems can be permanent/cooperative if there exists a unique
inner equilibrium [6]. Moreover
Result 1.3[4] The inhomogeneous hypercycle (I I I) is cooperative if
and only if there exists a fixed point in intSn.
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2. NONLINEAR GROWTH RATES ON 82
As a direct generalization of system (I I ) from a 3- simplex to a 2-
sphere is the following system of differential equation
(IV ) Xi = xi[qi + kuxi + ki2X~ + ki3X~ - ~]
±2 = x~[q2 + k2ixi + k22x~ + k23X~ - ~]
i:3 = x~[q3 + k3ixi + k32X~ + k33X~ - ~]
3 3
where <I> = L x¡(qi + L kiJx]). Here qi E R corresponds to the selfre-i=
i j=i
production or decay of the species i and kiJXJ represents the effect of
the sJ2ecies j on the reproduction of species i which is of mass action
type, "C'atalytic kiJ > O and inhibiting kiJ < O.
3. P ERMANENCE OF R EPLICATOR EQUATIONS ON 82 WITH
BOUNDARY
Definition 3.1: Let 8! = {x = (xi, X2, X3) E R3 : xi+ X~+ X~ =
c2, x3 > O} , a sphere of radius c. Let us take a horizontal section
by a hyperplane r = {x E 8! : X3 = k, k <e and e> O} in the upper
part of 8!. Then the 2- sphere with boundary is the subset of 8! obtained
by the above section together with small circle xi + x~ = c2 - k2,
x3 = k. It is denoted by G. In fact, G is a compact subset of R3 , where
the boundary of Gis given by B = {x E G: xi+ x~ = c2 - k2, x3 = k}.
For practica! purposes the positive half Gi of G is considered where
Gi = {x E G: Xi~ O,x2 ~ O,x3 = k}.
Now Gi has three boundaries namely;
Bi ={x E B: xi= vc2 - k2cose,x2 = vc2 - k2sine,x3 = k}
B2 ={x E B: X1 = O,x2 = ccose,x3 =e sine}
B3 ={x E B: Xi= ccose,x2 = O,x3 =e sine}
where o <e< 27r.
The boundaries B2 and B3 are two curves on the coordinate planes.
Then the system of differential equations defined on Gi is called a generalized
replicator system if it keeps the boundaries Bi, B2 and B3
of Gi invariant.
In this paper we are considering a special type of non-linear growth
rates which we call as inhomogeneous hypercycle on a 2- sphere
with boundary given as follows:
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(V) ±1 = (x1 - li)2[q1 + kl x§ + m1x5 - ~]
±2 = (x2 - l2)2[q2 + k2x5 + m2xi - ~]
X3 = (x3 - k) 2 [q3 + k3xi + m3x§ - ~]
where <I> = (x1 - li)2[k1x§ + m1x5J + (x2 - l2)2[k2x5 + m2xi] + (x3 -
k) 2 [k3xi + m3x§], ki, mi> O (i.e. mass action type catalytic);
k1, m1 = specific growth rate of x1 with respect to x2 and x3 respectively.
k2, m2 = specific growth rate of x2 with respect to x3 and x1 respectively.
k3 , m3 = specific growth rate of x3 with respect to x1 and x2 respectively.
qi = rate of selfreproduction or decay of the species i for i = 1, 2, 3 and
li = ../c2 - k2cose, l2 = ../c2 - k2sine is any point on B1 (O < e < 2?r)
and e is any finite positive number.
Theorem 3.1
Let a differential equation defined on the conÍpact set G1 of s¡ with
boundary B1 be given by (V). Then
(a) (V) is a generalized replicator system on G1 .
(b) If (V) is permanent then there exists a fixed point in intG1 with
respect to the boundary B1 . Conversely, if there exists a fixed point in
1 1 1 1 1 1
intG1 and - + - = - + - = - + - = A, where A is any arbi-m1
k3 m2 kl m3 k2
trary positive constant, then (V) is permanent.
Proof:
(a) On the boundary B1 of G1
(x1, X2, X3) = (li, l2, k) = (V ,...C..,2,,_..------..,,k-2=cose, VC2 - k2sine, k) , Ü < e < 21T.
Clearly from (V), it follows that ±1 =O, ±2 =O, ±3 =O. Thus (V) is a
replicator system of equation on G1 .
(b) Suppose (V) is permanent with respect to the boundary B1 . We
show that there exits an equilibrium point in intG1.
Let initially at T = T0 , x1(T0 ) > li , x2 (T0 ) > l2 , x3(T0 ) > k andas (V) is
permanent, so xi(t) > ó for sorne ó = min{l1, l2 , k} and all t suffi.ciently
large, i = 1, 2, 3, where x(t) = (x1 (t), x2(t) , x3 (t)) is the orbit of (V)
with (li, l2, k) = x, li = ../c2 - k2cose, l2 = ../c2 - k2sine (O< e < 2?r).
For T > T0 , let
1 {T
zi(T) =T lro x;(t)dt,i = 1, 2,3
and
w(T) = ~ Jio[(x1(t) - li)2{k1x§ (t) + m1x5 (t)}
+(x2(t) - l2)2{k2x5(t) + m2xi(t)} + (x3(t) - k)2[k3xi(t) + m3x~(t)]dt
The integral exists as xi( t) 2:: Zi, x~ ( t) 2:: l§ and k2 < x5 ( t) < c2 .
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Now zi(T) and W(T) are bounded as both are Riemann integrable. As
(V) is permanent, so there exists a compact set K in intG1 such that
the w- limit set of every orbit x(t) lies in K. So by compactness, there
exists a sequence Tn --+ oo and positive real numbers z~ , z;, z~ and '1!'
such that lim zi (Tn) = z~ and lim W(Tn) = W'.
3 3 3 3
Also, L x7 ( t) = c2 ::::} L zi (T) = c2 and hence L z: = c2 as L zi (Tn) = c2 .
i=l i=l i=l
Now
"1tl xi(t~ -li ] = -(xi~t1 )2 = -q1 -k1x~ -m1X~ + ~
::::} in flan ft[x1(~-l1]dt
= -ql - ~flan x~(t)dt - ~ Jion x~(t)dt + c'.L Jion <I>(x(t))dt
i=l
::::} in [x1 (T~)-li - xi(T~)-11 J = -q1 - ki z2 (Tn) - m1z3(Tn) + w(~n)
Since x 1 (Tn) is bounded (being in the compact set K) , it follows that
X1 (Tn) - li is also bounded. Hence Tn(xi(~n)-li) --+ O as n--+ oo so <loes
Tn(xi(~o)-li) --+ O (li = Vc2 - k2cos0) as n --+ oo. So, taking limit on
both sides as n --+ oo, we find
Similarly we get,
N ow let us consider,
ft [- x3 (i)-k] = (x3±_!k)2 = q3 + k3XI + m3X~ - ~
::::} A Jion ft[- x3(i)-k] dt =
q3 + ~ Jion xi(t)dt + ~ Jion x~(t)dt - c'k f1~n <I>(x(t))dt
::::} A [x3 (,fo)-k - x3 (T~)-k ] = q3 + k3 z1 (Tn) + m3z2 (Tn) - w (~n)
Since k < X3 (Tn) < e, it follows that - Tn (x3 (~n)-k) is bounded and hence
T,,(x3 (~n)-k) --+ O as n --t oo. Similarly, it follows that Tn(x3 (~o)-k) --+ O as
n --t OO.
So, taking limit on both sides as n--+ oo, we find
1 1 \]!'
Ü =q3 + k3Z1 + ffi3Z2 - -e
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Thus z:(i = 1, 2, 3) satisfies 3- linear equations
q1 + kiz; + m1z~ = q2 + k2z~ + m2z~ = q3 + k3z~ + m3z;
and
From above we find,
(1)
z~ ( m2 + m3 - k3) + z~ ( k2 + m3) + ( q2 - q3 - m3) = O ( 3)
Solving z~ and z~ from ( 2) and ( 3) and putting in ( 1), we find ( z~, z;, z~).
Consequently, (z~, z;, z~) is an equilibrium point of (V) in intG1.
We now find the sufficient condition of permanence of (V). As (V)
is a replicator system, so by Remark 1.1, it follows that t he sufficient
condition of permanence as given by Result 1.1 is also applicable for the
present replicator system (V). So we use the Result 1.1 in the present
case also to find out the condition of permanence.
Let p = (p1 ,p2 ,p3) be any fixed point in intG1 .
Then from (V)
Now
k 2 2 <I>(p)
q2 + 2P3 + m2P1 = -e
<I>(p) -- > max{q1, q2 , q3}
e
since ki > O, mi > O, i = 1, 2, 3.
Using (4) , (5) and (6) we get from (V),
41
(4)
(5)
(6)
(7)
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X. 1 = ( X1 - {i ) 2 [k l ( X22 - P22 ) + m1 ( X32 - p32 ) + -<P (p-) - -<P l
e e
. ( ) 2 [ ( 2 2) ( 2 2) <P (p) <P l X2 = X2 - l2 k2 X3 - P3 + m2 X1 - P1 + -- - -
e e
X. 3 = ( X3 - k )2[k3 ( X12 - P12 ) + m3 ( X22 - P22 ) + -<P(-p) - -<Pl
e e
where li = v'c2 - k2cose, l2 = ...;'c2 - k2 sine (O< e < 27r)
Now for any x(t) on G1, x1 (t) > li, x2(t) > l2 , x3(t) > k.
1 1 1
We now construct P(x) = e-m1k1(x1-l1le-m2k2(x2-t2le-m3k3(x3-k)
Clearly P satisfies
(i) P(x) = O
(ii) P(x) > O
Now P(x) = _1_ x + _1_ ±2 + _1_ ±3
P(x) m1k1 (x1-li)2 m2k2 (x2-l2)2 m3k3 (x3-k)2
= _1_[k (x2 -p2) + m (x2 _ p2) + <I>(p) _ ~]
m 1k1 1 2 2 1 3 3 e e
+-1-[k (x2 _ p2) + m (x2 _ p2) + <I>(p) _ ~] m2 k2 2 3 3 2 1 1 e e
+-1-[k (x2 _ p2) + m (x2 _ p2) + <I>(p) _ ~]
m3 k3 3 1 1 3 2 2 e e
= (x22 _ p22 )(m-11 + ..l.)+ (x2 _ p2)(...l. + _1 ) + (x2 -p2)(...l. + _1 ) k3 3 3 k1 m2 1 1 k2 m3
+(m:k3 + m~k1 + m~k)(<I>~) - ~)
- (-1- + _1_ + _1_) ( <l>(p) - ~)
- m3k3 m1k1 m2k2 e e
1 1 1 1 1 1
only when - + -k = - + - = - + - = A where A is any arbi-m1
3 m2 kl m3 k2
trary positive constant and since (p1,p2,p3) is a fixed point in intG1 ,
xi + x~ + x5 = Pi + P~ + P5 = c2.
Thus
P(x) _ (-1- + _1_ + _ 1_ )(<1>(p) _ ~)
P(x) - m3k3 m1k1 m2k2 e e
1 1 1 1 1 1
provided - + -k = - + -k = - + -k = A where A is any arbitrary
m1 ~3 m2 ~1 m 3 2
positive constant; which is continuous in G1 .
Hence we can write ~~:~ = w(x) (with standard notation).
Again w(x) > (m:ka + m~ki + m~k2 )[max{q1,Q2 , q3}- ~] by (7).
Now we want to apply the sufficient condition of Result 1.1 for our
system (V). First we show that for the system (V) , Result 1.1 reduces
to showing that for each x E B1 and every E > o (arbitrarily small),
there exists sorne T > T0 such that
1 {T
T lro w(x(t) )dt > O
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Now ~Ir~ '11 (x(t)) dt > O
if ~Ji'a(m~ki + m~k2 + m~k3 )[max{q1,q2,q3}- <I>(:(t))]dt > O
i.e. if
1 JT -T <I>(x(t))dt < max {q1, q2 , q3 } +E
e To
sm. ce -k1 - + -k1 - + -k1 - > o. m1 ·1 m2 2 m3 3
(8)
Obviously if (8) is true, then by sufficiency it fallows that the system is
permanent. Therefare it remains to prove that the condition (8) holds.
In fact , suppose the condition <loes not hold. Then far any x in B1 ,
lT ¡r <I>(x(t))dt ~ max{q1,q2 ,q3 } +E far all T and far all E(> O) , how-c
}~ .
ever large they may be.
Now (v'c2 - k 2cosB, .../~c2~-~k~2 sin0 , k) E B1, but x(t) E G1, so if x (t) =
(x1(t) , x2 (t) , x3(t)), then x1(t) > li, x2(t) > l2, x3 (t) > k, where
li = v'c2 - k2cosB , l2 = v'c2 - k2sin0 (O < e< 2n). Now
:i:3(t) - + k 2 + 2 <I>
(x3 (t)-k)2 - q3 3X1 ffi3X2 - e
=? ~ Ji'o (x3(:;(~k)2 = q3 + * Ji'o xi(t)dt +y Ji'o x§(t)dt - e~ Ji'o <I>(x (t))dt
=? ~[x3(io)-k - x3(i)-k]
:-::;; q3 + * Ji'o xi(t)dt +y Ji'o x§ (t)dt - max{ q1 , q2, q3} - E
=? _ _!_ 1 <
T (x3(T)-k) -
q3 + * Ji'o xi(t)dt +y Ji'o x§(t)dt - max{ q1 , q2, q3} - E - ~ (x3(io)-k)
< * Ji'o xi(t)dt +y Ji'o x~(t)dt - E
[q = q3 - max{q1, q2, q3} :-::;; O,~ (x3(A- k) > O]
= -ó(ó > O) (say) , by taking E(> O) arbitrarily large.
So, - ~ x:3(i)-k < -ó
1 1 .r
=? T X3(T)-k > u
=? T[x3(T) - k] < i
=? x3(T) - k < 8~ where A -* O as T -* oo.
Hence as T - oo, x3(T) -* k.
Thus, if the sufficient conditions of permanence far the system (V)
is violated, then from above it fallows that any solution of generalized
replicator system on G1 shows a tendency of extinction. Therefare, we
must assume that the condition (8) holds. Therefare (V) is permanent
with respect to the boundary B1 . Hence existence of fixed point in intG1
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1 1 1 1 1 1
and - + -k = - + -k = - + -k = A are sufficient conditions for
m1 3 m2 i m3 2
permanence of (V) with respect to the boundary B1.
This completes the proof of the theorem. D
Remark 3.1: Suppose the differential equations defined on the compact
set G1 of S! with boundaries B2 and B3 be respectively given
by:
(VI) ±1 = xi[q1 + k1x§ + m1x~ - ~]
±2 = (x2 - c cose)2[q2 + k2x~ + m2xi - ~]
X3 = (x3 - c sine)2[q3 + k3xi + m3x§ - ~]
where 1> = xi[k1x§ + m1x~] + (x2 - C cose) 2 [k2X~ + m2xij + (X3 -
c sine)2[k3xi + m3x§], ki > O, mi > O and (O, c cose, c sine) is any
point on B2 (O< e< 2n) ande is any finite positive number.
and
(VII) ±1 = (x1 - c cose)2[q1 + k1x§ + m1 x~ - ~]
±2 = x§[q2 + k2x~ + m2xf - ~]
X3 = (x3 - c sine)2[q3 + k3xf + m3x§ - ~]
where 1> = (x1 - C cose)2[k1x§ + m1x~] + x§[k2X~ + m2xf] + (X3 -
c sine)2[k3xi + m3x§], ki > O, mi > O and (c cose, O, c sine) is any
point on B3 (O< e< 2n) ande is any finite positive number, then
also the conclusion of Theorem 3.1 remains true.
Discussion:
1. On concentration simplex, an inhomogeneous hypercycle is permanent
if and only if there exists a fixed point in the interior of the simplex
[Result 1.3]. Similarly in case of an inhomogeneous hypercycle on a 2-
sphere with boundary the necessary condition of permanence remains
the same; this means if (V) is permanent then there exists a fixed point
in intG1. The main difference in the said two cases arises only in the sufficient
condition of permanence viz., that (V) is permanent if there exists
a fixed point in intG1 and the condition -1 + k1 = -1 + k1 = -1 + k1 = A m1 3 m2 1 m 3 2
holds, where A is any arbitrary positive constant. This condition means
that the sum of the reciprocals of catalytic growth coefficients of species
1, 3, 2, 1 or 3, 2 taken in pair remains equal to the same constant.
2. Comparing (I I I) with (V) it is noted that in (V) the species 1
acts catalytically on 2 and 3, 2 acts catalytically on 3 and 1, 3 acts on
1 and 2 respectively in a inhomogeneous hypercycle on a 2-sphere with
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boundary. But in the inhomogeneous hypercycle ( I I I ) on 3-simplex it
is seen that the species 1 acts catalytically on 2, 2 acts catalytically on
3 and 3 on 1 respectively. Thus it appears that for (V) , the catalytic
action of an individual species on the rest is comparatively complex than
the corresponding catalytic action of an individual species on the rest
in case of ( I I !) . This complex phenomenon can be explained if we note
that for (V) the growth rate is expressible as a separable biquadratic
one, whereas the corresponding growth rate in (I I !) is expressible as
a separable quadratic one. This is an interesting achievement while
generalizing system (I I !) to system (V).
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