FEEDBACK REGULATION OF LOGISTIC GROWTH

Sufficient conditions are obtained for the global asymptotic stability of the positive equilibrium of a regulated logistic growth with a delay in the state feedback of the control modelled by


Introduction
The autonomous ordinary differential equation rN(t) [1 N(t)  has been a basis for the development of several models of dynamics of populations in mathe- matical ecology.It is an elementary fact that if N(0) > 0 then the corresponding solution of (1.1) satisfies lim N(t)= K 1.2 and the convergence in (1.2) is monotonic.It has been found that such a monotonic con- vergence to the equilibrium is not realistic in certain laboratory populations in temporally uniform environments (Hutchinson [9]); also some of the negative feedback effects (such as the accumulation of toxic residuals etc.) act with some time delay (Volterra [13]).One of the possible ways of modifying (1.1) so as to model these additional features is to consider a time lagged equation of the form d N ( t ) d t rN(t)[1-(aIN(t) +a2N(t- 1.3 where a, a2, K, r (0, oo).Equations of the form (1.3) have been discussed by several authors in the literature on population dynamics.
Along with (1.3) one considers an initial condition of the form N() () > 0; (0) > 0; e C([-,0I, m+). 1.4 Solutions of (1.3) and (1.4) satisfy [f0'{ (N() + N(,-))}] t>0 N(t) N(0)exp r 1-K from which one can see that solutions of (1.3) and (1.4) are defined for all > 0 and also satisfy g(t) > 0 for all > 0. It is known that if al > a2 then all solutions of (1.3)-(1.4)and the convergence in (1.5) is unconditional on the size of delay for details see Lenhart and  Travis [12] or Gopalsamy [4]).We shall suppose that we have a situation where the equilibrium N* of (1.5) is not the desirable one (or affordable) and a smaller value of N* is required; thus we are required to alter the system (1.3) structurally so as to make the population stabilise at value lower than that in (1.5).One of the methods of accomplishing this is to introduce a "feedback" control variable and this can be implemented by means a biological control or some harvesting procedure.We formulate one such model below.Stability of feedback control systems has been discussed in the books by LaSalle and Lefschetz [10], Lefschetz [11] and Aizerman and Gantmacher [1].
The controlled system (2.1) has a positive equilibrium (n*, u*) defined by The following preparation will be useful in the proof of our result on the global attractivity of (', =*).

2.13
where B, fl d e positive nbers to be lected below suitably.Note that by the deflation ff in (2.10), V is nonnegative.CMculating the rate of ge of V Mong the solutions of (2.7)-(2.10), #ar 1 2(o(t_ r)).

2
where fl is y positive number; we obrve that it is sufficient to ch B to be the positive rt of the quadratic equation For this choice of B, we have from (2.17), dV 2.20 It follows from (2.21) that x E Ll(0, oo).We want to verify that x is uniformly continuous on (0, oo) and for this it is sufficient that k is uniformly bounded on (0, oo).From (2.7) we can see that k will be uniformly bounded if both x is uniformly bounded and a is uniformly bounded above.The uniform boundedness of x is immediate from (2.21) since (2.21) implies Suppose lim sup a( then there exists a sequence {tin } OO as moo such that do" >_ 0; a(t,, r) _< a(t,,), a(t,,,) oo as m oo.

2.24
From (2.24) and the uniform boundedness of x we can conclude that is uniformly bounded with the implication that x is uniformly continuous on (0, oo).It is easy to infer from (2.21) that x 2 E LI(0, oo); by Barbalat's lemma (see Corduneanu [3]) we conclude that lim x(t) 0.
Since x(t) 0 as oo, it will then follow from (2.26) that 2.26 lim a(t) 0 and lim y(t) 0.

2.28
This completes the proof.

Stabilisation and bifurcation control
It is known that the positive equilibrium of the delay logistic equation becomes linearly unstable when rr > and for small positive (rr-) there exists a periodic solution solution of (3.1) bifurcating from the positive equilibrium (Hassard et al. [8]).We suppose that it is required to stabilise the system (3.1)possibly at a different equilibrium and thereby avoid the bifurcation to periodicity.Accordingly we consider the feedback control where r,K,c,a, b E (0, x), r [0, x) and (3.2) is supplemented with initial conditions of the type (2.1).The system (3.2) has a positive equilibrium (n*, u*) where u*=-+-- In terms of the variables x, y, a defined in (2.6) we rewrite the system (3.2) as follows: dz(t-----2) -az(t) + b(a(t)); (a(t)) = n'[e " (') 11 ] dt dtr(t) -crz(t) r dt -ff (( ). 3.4 Theorem 3.1.Assume that the patmaneters of the system (3.2) satisfy bc 1 > and a > (1 + vf)r.

3.5
Then the positive equilibrium (n*, u*) of (3.2) is linearly asymptotically stable irrespective of the size of the delay r.
Proof, The equilibrium (n*, u*) of (3.2) is linearly asymptotically stable if the trivial solution of the system (3.4) is linearly asymptotically stable.Such a linear system is A2+A a+-n e x +e +rn'=O.

3.7
If r 0, (3.7) becomes A+A a+n + + =0 who roots have negative re pts.Thus the tfivi lution of the line system (3.6) is ymptoticMly stable when r 0. By the continuity of the roots of (3.7) on the parameters (for details see Cooke d Grossm [2]), it will follow that M1 the roots of (3.7) will have negative re pts for sml positive ues of r.We wt to find scient conditions so that the reM parts of the roots of (3.7) will have negative re pts whatever the size of the delay r.
We note first A 0 is not a root of (3.7).If the root of (3.7) have zero or positive ral parts, then for some positive r, A +iw, w > 0 are roots of (3.7).If we can show that , +iw cannot be roots of (3.7), then it will follow that all roots of (3.7) lie on the left half of the complex plane for all r > O.For instance if A +iw is a root of (3.7) then -w + iw a + -n e + -n e iwr +bcrn* 0.

3.8
Separating the real and imaginary parts of (3.8), w bcrn" w-n* sin wr + a n cs wr } r ar 3.9 -aw w-n cos wr-n sin w7.

3.10
It can be found that if (bcrn*) > --n 3.11 a 2 > 2bcrn* + 3.12 then (3.10) will have no positive w 2 and hence (3.7) cannot have a pure imaginary root.By the first of (3.5), (3.11) is satisfied.We note from the second of (3.5) and the fact bcn* < a and n* < K 3.14 and hence (3.12) holds.Thus (3.10) has no real roots; it will then follow that (3.7) cannot have pure imaginary roots for any positive r.We can conclude that a delay induced switching from stability to instablity cannot take place.Thus the absolute (or delay independent) linear stability of the positive equilibrium of (3.2) follows.This completes the proof.
We remark that without the indirect control u in (3.2), the system (3.1) can have periodic solutions arising through a Hopf-type bifurcation for suitable delays.We have illustrated that an appropriate indirect feedback control can be used to avoid the occurance of a Hopf-type bifurcation and stabilise a logistic growth by structurally altering the logistic growth. 4. Feedback control with time delay In this section we continue with a discussion of the dynamics of the system (3.1)together with a feedback control when there is a delay in the state feedback of the control variable.We shall consider the following system dn(t) dt du(t) dt -au(t) + Im(t-r) 4.1 in which the feedback to the stabilising control u involves a time lag r.We assume r, a, b, K, r are positive numbers.One can see that (4.1) has a nontrivial steady state (n', u*) where aK bK n* u* 4.2 a + Kbc; a + Kbc" We shall assume that the positive numbers r, K, a, b, c, r are such that bcK er r < 1.
For convenience in the following we define a as follows: 1 (bK + ).
The following Lemma provides a priori estimates of the solutions of (4.1) and (4.6).
Lemma 4.1 Let be a positive number such that (4.4) holds and let (n(t), u(t)) denote arbitrary positive solution of (4.1) and (4.6).Then there exists a T > 0 such that e <_n(t) <_ L m _<u(t) _< M /'or > T*; (If necessar choose smMl enoush so that m > 0). 4.8 Proof.By the positivity of u, we have from (4.1) that dn(t) { n(t-r)} dt _<rn(t) 1-K for t>0.4.9 There are two possibilities; n is oscillatory about K or n is nonoscillatory about K.If n(t) is not oscillatory about K then there exists a To such that either n(t)>K for t>To or n(t)<K for t>To.

4.10
If the second alterative holds then we have n(t) <_ Ke for > To.
Suppose n(t) > K for > To; then dn(t) dt <0, for t>T,+r and hence for some no > 0, we have n --, no, as oo.
One can show that no < K (the details of proof omitted) from which it will follow that there exists a To such that n(t) < Ke for > To.
Let us now suppose n is oscillatory about K and let n(t*) denote an arbitrary local maximum of n; then and therefore d(t') < .(t')[X.(t'-) 0= d--q--U n(tr) < K; 4.12 a consequence of (4.12) is that there exists E It* r,t*] such that n() K. Integrating the inequality (4.9) from to t*, we have which implies that gn---) <_ r 1-K <_ r ds < r ds rr n(t) <_ Kerr.
Since n(t*) is an arbitrary local maximum of N, we can conclude that n(t) < n(t*) < Kerr= L

4.13
for _> To, where To is the first zero of the oscillatory n.We have from (4.13) and the second of (4.1) that du(t) d"'-<--au(t) + bL for > T, + r.

4.14
We can compare solutions of (4.14) with those of dx,:.tt-ax(t) + bL, x(T, + r) u(To + r).It is easy to see that solutions of (4.15) satisfy bL Thus by comparison of (4.14) and (4.15) we can conclude that u(t) < x(t) for > T + r.
Thus there exists a T > 0 such that bL u(t) < ---+ e M for > T.

a
The priori upper bounds of n and u in (4.7) follow from (4.13) and (4.16).Using these upper bounds of n and u one can derive the lower bounds in a similar way.We shall omit the details.
The next result provides verifiable sufficient conditions for the global asymptotic stability of the positive equilibrium (n*, u*) of (4.1).

4.45
It is now a consequence of (4.45) that (4.38) has a unique positive root.Thus t* n L" and hence M* u* m*.Combining (4.30) and (4.31), we obtain and this completes the proof.