We have seen in Theorem 4.1.1 that any solution to the nonlinear equation (4.8) converges to a steady state. Can this result be extended to Equation (4.10) where the death rate is also nonlinear ? The result in Theorem 4.1.4 answer negatively to this question. Indeed it ensures the existence of functionsf and g,and parameterspandqsuch that Equation (4.10) admits periodic solutions. Here we give examples of such functions and parameters and more precisely we prove, thanks to the Poincar´e-Bendixon theorem, that any solution with an initial distribution in the eigenmanifoldEwhich is not a steady state converges to a periodic solution. Then we extend this result by surrounding this set of initial distributions by an open neighbourhood inH.
In the proof, we need to know the dependency of some quantities on the parameterspandq. Since we do not know the dependencies ofMp´
xpUpxqdxonp,we consider an equation slightly different from (4.10), namely
B
Btupt, xqf
´
xpupt, xq
´ xpUpxq
B
Bx xupt, xq
g
´
xqupt, xq
´ xqUpxq
upt, xq Fγupt, xq. (4.56) Clearly the existence of functions f and g for which persistent oscillations appear in Equation (4.56) ensures the same result for Equation (4.10) (up to a dilation off andg). Now let set the assumptions on the two functionf andgwhich allow to obtain periodic oscillations. We consider differentiable increasing functionsf andgsuch that
fp0q¡gp0q0 and fp8q gp8q8. (4.57) Moreover we assume thatg is invertible and define onR the function
ψpWq:f
Wkppqqg1pWq . (4.58)
To ensure the existence and uniqueness of a nontrivial equilibrium, we assume that
D!W8
¥0, ψpW8
qW8 and moreover ψ1pW8
q 1. (4.59)
This steady state is unstable if, denotingQ8:W8kqg1pW8q,we have Q8 pWkp
8
f1pWkp
8
Q8
qWkq
8
g1pWkq
8
Q8 q
W8
k ¡0. (4.60)
Then solutions to Equation (4.56) with initial distribution close to the set EztQ8UpW8;qu exhibit asymptotically periodic behaviors. More precisely, we have the following result
Theorem 4.4.1. Consider increasing differentiable functionsf andgsatisfying conditions (4.57),(4.59) and (4.60), a parameter γPp0,2sand a fragmentation kernelκwhich satisfies Assumption (4.33). Then there exists an open neighbourhood V of EztQ8
UpW8;qu in H such that, for any initial distribution u0PV,there exist periodic functions Wptq andQptqsuch that
}upt,qQptqUpWptq;q}HÝÝÝÑ
tÑ8 0. (4.61)
Before proving this theorem, we give examples of functionsf andgsuch that conditions (4.57), (4.59) and (4.60) are satisfied. These examples, together with Theorem 4.4.1, give the proof of Theorem 4.1.4.
Example 1. Assume that there existsC¡0 such that for allx¥0, gpxq¤C xg1pxq, thenψpWqW has a unique solution forkpqpq¡C.Indeed, if we compute the derivative ofψwe find
ψ1pWqWkppqq
kppqqg1pWq
W pg1q1pWq
and gpxq¤C xg1pxqimplies that xpg1q1pxq¤C g1pxq. So ifkpqpq¡C, ψ decreases and Assump-tion (4.59) is fulfilled. If moreoverfp1qgp1q1, then the unique nontrivial equilibrium is given by W8
1.Then condition (4.60) is satisfied forp¡ g
1
p1q 1k f1p1q .
Example 2. Consider the case gpxq x and p q, and assume that fpxqxhas a unique root x0
andf1px0q1 0.Then ψpWqfpWqand Assumption (4.59) is satisfied. Moreover, condition (4.60) writespf1pW8q¡1 1k so it is satisfied for plarge enough.
Now we give a lemma useful for the proof of Theorem 4.4.1.
Lemma 4.4.2. Consider a dynamical system inRn with a parameter εptq
X9 FpX;εptqq (4.62)
with F P CpRnRnq. Assume that for any vanishing parameter }εptq} ÝtÝÑ8ÝÑ 0 the solutions to Equa-tion (4.62) are bounded. Then for any solution Xε associated to }εptq} ÝtÝÑ8ÝÑ0, there exists a solution X0 associated toε0 such that Xε andX0 have the sameωlimit set.
Proof of Lemma 4.4.2. Let Xptq a solution to System (4.62) with }εptq} Ñ 0. By assumption, Xptq is bounded, soX9ptqis also bounded since F is continuous. Now consider a sequencettkukPN wich tends to the infinity and define the sequencetXkpqubyXkptqXpt tkq.This sequence is bounded inW1,8pR q
so there exists a subsequence which converge to X8pq. This limit is a solution to Equation (4.62) with ε0.We takeX0:X8 that ends the proof of Lemma 4.4.2.
Proof of Theorem 4.4.1. We divide the proof in two parts: first the result foru0PEand then the existence of a neighbourhoodV ofE inHwhere the result persists.
First step: u0PE.
Foru0PEzt0u,there areW0¡0 andQ0¡0 such that u0pxqQ0UpW0;xq.
Then, ifupt, xqis the solution to Equation (4.56) andW is the solution to W9 W
k
f
Mprusptq MprUs
W
withWp0qW0,the relation holds for allt¡0 andx¡0 upt, xqQptqUpWptq;xq
whereQptq:Q0e´0tpWpsqgpMqruspsq{MqrUsqqds.Then we can compute Mprusptq
ˆ 8
0
xpupt, xqdxWkpptqQptqMprUs
and finally we obtain the reduced system of ODEs satisfied bypW, Qq
$
We prove that System (4.63) has bounded solutions and a unique positive steady state which is unstable.
Then we use the Poincar´e-Bendixon theorem to ensure the convergence to a limit cycle.
The fact that 0 fp0q¤f ¤fp8q 8and thatgincreases from 0 to the8ensures that the solution remains bounded. LetpW8, Q8qa positive steady state. It satisfies
W8
f WkpQ
g WkqQ
and so, sincegis invertible,Q8
W8kqg1pW8
q.ThenW8 is solution to the equation W8
but Assumption (4.59) ensures the uniqueness of such a solution. Now look at the stability of this positive steady state. We write system (4.63) under the form
The trace of this matrix is
TQ8 pW8kpf1pW8kpQ8qW8kqg1pW8kqQ8q
Thus whenT ¡0,namely when Assumption (4.60) is satisfied, the two eigenvalues have positive real parts and the positive steady state is unstable. Now we prove thatpW, Qqremains away from the boundaries ofpR q
2.For this we write that
t¡0, W :minpW0, fp0qq¤Wptq¤maxpW0, fp8qq:W
and then
Q¥minpQ0, Wkpg1pWqq.
Sincefp0q¡0, W ¡0 forW0¡0 so any solution withW0¡0 andQ0¡0 stays at positive distance of the boundaries ofpR q
2.Then the Poincar´e-Bendixon theorem (see [68] for instance) ensures any solution to System (4.63) withW0¡0, Q0¡0 andpW0, Q0qpW8, Q8
qconverges to a limit cycle.
Second step: Existence ofV.
Letu00 inHand define fromupt, xqsolution to Equation (4.56) with initial distributionu0a function v by
vphptq, xqWkptqupt, Wkptqxqe´0tpgpMqruspsq{MqrUsqWpsqqds withW solution to
W9 W k
f
Mprus MprUs
W
and h solution to h9 W with hp0q 0. We have already seen in Section 4.2.3 thath : R Ñ R is one to one sincehptq¥klnp1 ktq. We takeWp0q1 to havevpt0,qupt0,qu0.Thanks to Theorem 4.2.1 we know thatv is solution to
Btvpt, xq Bx x vpt, xq
vpt, xqFγvpt, xq and the GRE ensures the convergence
vpt, xqÝÝÝÑ
tÑ8
ˆ
φpxqu0pxqdx
Upxq. As a consequence we have the equivalences, for anyp¥0,
Mprusptq
tÑ8̺0MprUsWkpe´0tpWpsqgpMqruspsq{MqrUsqds
so, if we defineQptq:̺0e´0tpWpsqgpMqruspsq{MqrUsqds,we find that the reduced system (4.63) is “asymp-totically equivalent” to Equation (4.56). More precisely we define as in the proof of Theorem 4.3.1
εpptq Mprusptq
MprUsWkpptqQptq1 we have thatεpÑ0 andpW, Qqis solution to
$
&
%
W9 W
k f p1 εpqWkpQ
W
, Q9 Q W g p1 εqqWkqQ
.
(4.64)
Now we prove that if W0 and Q0 are positive and pW0, Q0q pW8, Q8
q, then if }u0Q0UpW0;q}H is small enough, the solution u to Equation (4.56) converges to a periodic solution. Denote by d the distance betweenpW0, Q0qand pW8, Q8
q. SincepW8, Q8
q is a source for System (4.63), there exists a ball with radiusρ dsuch that the flux is outgoing, namely
pW, QqPBBppW8, Q8
q, ρq, FpW, Qqn¡0 (4.65) where n is the outgoing normal of BppW8, Q8
q, ρq. Then if we define by FpW, Q;εp, εqq the flux of Equation (4.64), we have by continuity off andg that there existsε0 such that (4.65) remains true for FpW, Q;εp, εqqprovided thatεp andεq stay less thanε0.But we know from the proof of Theorem 4.3.1 that there exists a constant Cp ¡ 0 such that for all time t ¡ 0, εp ¤ Cp}u0Q0UpW0;q}. So for
}u0Q0UpW0;q}¤ Cε0
p Cq,the solution to System (4.64) cannot converge to the positive steady state
pW8, Q8q. Thanks to the same arguments, if }u0Q0UpW0;q} is small enough, then pW, Qq remains
away from the boundaries ofpR q
2.We obtain thanks to Lemma 4.4.2 that for}u0Q0UpW0;q}small enough,pWptq, Qptqqconverges to a limit cyclepWptq,Qrptqq.Then we write
}uQrUpW;q}¤}uQUpW;q} }QUpW;qQrUpW;q}
and we conclude as in the proof of Theorem 4.3.1 that the solutionuto Equation 4.10 converges inHto QrUp
W;q.Finally we have proved, for anypW0, Q0qPpRq
2
ztpW8, Q8
qu,the existence of a ball centered inQ0UpW0;qsuch that any solution to Equation (4.56) with an initial distribution in this ball converges to a periodic solution. Then Theorem 4.4.1 is proved forV the reunion of all these balls.
To illustrate the convergence to a periodic solution for solutions to Equation (4.56), we plot in Fig-ure 4.2 a solution to Equation (4.63) with an initial condition close to the steady statepW8, Q8qand for coefficients which satisfy the assumptions of Theorem 4.4.1.
0.7 0.8 0.9 1 1.1 1.2 1.3 1.4
0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6
W
Q
0 10 20 30 40 50
0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6
t
W(t) Q(t)
Figure 4.2: A solution to System (4.63) is plotted in the phase planpW, Qq(left) and as a function of the time (right). The coefficients areγ1, p2, q5, fpxq1 e1ex4 andgpxq0.9x.