Pattern formation of a Schnakenberg-type plant root hair initiation model

This paper concentrates on the diversity of patterns in a quite general Schnakenberg-type model. We discuss existence and nonexistence of nonconstant positive steady state solutions as well as their bounds. By means of investigating Turing, steady state and Hopf bifurcations, pattern formation, including Turing patterns, nonconstant spatial patterns or time periodic orbits, is shown. Also, the global dynamics analysis is carried out.


Introduction
Reaction-diffusion systems have definitely become a powerful tool for explaining biochemical reactions and species diversity because of the incorporation of elements including interaction mechanism and spatiotemporal behavior.In this paper, our attention is paid to the following spatially homogeneous plant root hair initiation model proposed in [20] which is viewed as the generalisation of Schnakenberg system [25] where all parameters are positive and Ω ∈ R n is a bounded domain.From the perspective of biology, initiation and growth of root hair (RH) result from the accumulation of active small G-proteins ROPs (Rhos of plants).In fact, the active ROPs are derived both from the transformation of inactive ROP by guanine nucleotide exchange factors (GEF) and from the induction of auxins together with other substances.Based on the mechanism above, the model simulates the interactions between inactive and active ROP (the detailed modeling process is found in [1,20]).u(x, t) and v(x, t) in (1.1) indicate concentrations of active and inactive ROP, respectively.k 1 + k 2 u 2 is the rate of ROP activation, c is the unbinding rate of active ROP, r shows the removing rate of active ROP by degradation, recycling, or other irreversible binding, and the inactive ROP is produced at rate b.
The extremely general model to include cases above is just the same as system (1.1), and we will continue to treat its patterns on the basis of previous extensive works.Our paper aims at pattern formation in the system (1.1).To explore existence and nonexistence of pattern formation, it is essential to discuss problems about steady states.In detail, by analyzing characteristic equation as well as some classical techniques (including comparison theorem, lower-upper solutions, priori estimate), constant bounds, existence and uniqueness of solutions in parabolic equation (1.1) are determined, also, another points are local and global asymptotically stability of constant equilibrium.Moreover, equiped with priori bounds, energy estimates and Leray-Schauder degree theory in elliptic partial differential equations (PDEs) we prove existence together with nonexistence of nonconstant positive steady states, which explains whether system (1.1) processes spatial patterns.Moreover, by taking global dynamics of PDE system into consideration, the diversity of patterns is revealed.In detail, analysis for bifurcations indicates Turing, nonconstant spatial as well as time-periodic patterns.

Stability of equilibrium 2.1 Local stability
Obviously, we are able to find that system (1.1) has a unique equilibrium The locally asymptotical stability of E can be analyzed.
Proof.Initially, the linear operator at E is implying a sequence of matrices where Assume λ is the eigenvalue of L, and the characteristic equation is written as Next, it is essential to discuss the eigenvalues of (2.3) because all eigenvalues with negative real parts demonstrate that E is locally asymptotically stable, otherwise E is unstable.
for tr(L i ) < 0 and det(L i ) > 0, that is, the equilibrium is stable.By some calculation, the condition is equivalent to then tr(L 0 ) > 0 causes at least one eigenvalue with positive real part.As a result, we have an unstable equilibrium.

Global stability
The main conclusion about global stability of E in this subsection is demonstrated as follows.
Theorem 2.2.Suppose that the domain Ω ⊂ R n is bounded and the boundary ∂Ω is smooth.
Proof.(i) Follow the marks in [19] and denote Apparently, (1.1) is a nonquasimonotone system.Let ( û, v) = ( ū(t), 0) and ( ũ, ṽ) = (u * , min{v * , v(t)}), where and (2.7) Subsequently, we are dedicated to proving that ( û, v) and ( ũ, ṽ) are lower and supper solutions of (1.1), respectively.In fact, and It is also easy to check the boundary-initial conditions are satisfied, so a pair of lower and upper solutions is definitely found.
(ii) About the global stability of (u * , v * ), the second equation of system (1.1) admits that Thus, Lemma A.1 in [39] and comparison principle show that lim sup This yields that there exists a constant T ε Now considering the first equation in (1.1), it is easy to conclude that for x ∈ Ω and t ≥ T ε 1 , The roots of ζ 1 (u) then are u ε 1 and u ε 2 , where and < 0 and again apply Lemma A.1 in [39] and comparison principle to get lim sup Let ε → 0, and Lemma A.1 in [39] together with the second equation of (1.1) give us that lim inf Thus, it is obtained that for 0 < ε < u 1 , there exists and Obviously, ϕ, ψ are decreasing and increasing, respectively.ūi , vi , such that Applying the monotonicity of ϕ and ψ and the relationship above, it follows Based on the monotonicity of sequences, assume that (2.17) Plugging the functions into the equations, then it should be that Combining the first and last two equations, respectively, gives us that Consider the equations above together, it follows that

Existence and nonexistence of nonconstant positive steady states
In this section, we investigate whether there exist nonconstant positive steady states for system (1.1).In other words, the solutions of (1.2) should be considered.

Nonexistence of nonconstant positive steady states
In the beginning, we focus on the priori estimate of positive solutions for (1.2).According to Proposition 2.2 in [15] and Theorem 8.18 in [9] (also see [13]), the following conclusion is demonstrated.Proof.Let (u, v) is a solution of (1.2).First integrating both sides of (1.2) by parts gives that Adding the two equalities above, we obtain that According to the first equation, the following relationship is satisfied Thus, Theorem 8.18 in [9] shows that there exists a positive constant C such that Adding the equations in (1.2), denoting w = D 1 u + D 2 v and w(x 0 ) = max Ω w(x), then Applying Proposition 2.2 in [15] yields that b − ru(x 0 ) ≥ 0, that is, u(x 0 ) ≤ b r .As a result, we finally get that Next, we discuss the priori estimate of v(x).As a detail, set v(x 1 ) = max Ω v(x) and v(x 2 ) = min Ω v(x).
Then it follows from Proposition 2.2 in [15] that and it is easy to see that Again because of Proposition 2.2 in [15], Therefore, it is concluded that With the help of Theorem 3.1 and methods together with results in [31], the nonexistence of nonconstant solutions is stated.
|Ω| Ω udx, and v 0 = 1 |Ω| Ω vdx.Consequently, u 0 = b r from (3.2), and Multiplying the first equation by u − u 0 and using the integration by parts in Ω, we have ) ) (3.8) In the same way, we can also get (3.9) In addition, based on the Poincaré inequality where µ 1 is the smallest positive eigenvalue of −∆.The results above lead to where The previous works [22,29] imply that β is an eigenvalue of D U Γ(U * ) on X i if and only if where m(µ i ) is the algebraic multiplicity of µ i .Obviously, when H(µ) = 0 has two different positive roots µ ± with µ + > µ − .Thus, H(µ) < 0 if and only if µ ∈ (µ − , µ + ).
Consequently, the following result describing the existence of nonconstant steady states of (1.2) is derived.
2) has at least one nonconstant solution in Λ.
Proof.Define a mapping Ĥ : where D is defined in Theorem 3.2.
It is easy to obtain that solving (1.2) is equivalent to finding the fixed points of Ĥ(•, 1) in Λ.From the definitions of D and Λ, we easily get that Ĥ(•, 0) has the only fixed point (u * , v * ) in Λ.
On the one hand, we deduce that Suppose that (1.2) has no other solutions except the constant one (u * , v * ), then On the other hand, from the homotopic invariance of Leray-Schauder degree, it is reasonable that leading to a contradiction.Therefore, this shows that there exists at least one nonconstant solution of (1.2).
Corollary 3.4.If K > 0, µ j < c+r KD 1 < µ j+1 for some integer j ≥ 1 and ∑ j k=1 m(µ k ) is odd, where K is defined in Theorem 2.1.Then there exists a large positive number D * such that (1.2) has at least one nonconstant solution as D 2 > D * .
Proof.Looking for the explicit expression of H(µ), it is clear that (3.14) holds for sufficient large D 2 .Besides, K > 0 gives that As a result, i = 1 in Theorem 3.3 implies this corollary.

Bifurcation analysis
In order to better understand patterns of system (1.1), we consider bifurcations from the positive constant equilibrium, such as Turing, steady state and Hopf bifurcations.

Turing bifurcation
Several theorems could answer the existence of Turing bifurcation.In this section, we still employ quantity K in Theorem 2.1 to give our results.
Proof.Once more, we study the characteristic equation (2.3).First, without diffusion term, sufficient conditions for locally asymptotically stable equilibrium E in ordinary differential equation (ODE) are tr(L 0 ) < 0 and det(L 0 ) > 0, which is equivalent to If the condition (H 1 ) is satisfied, then for all i ≥ 0, tr(L i ) < 0. So as long as ∃ i ∈ N such that det(L i ) < 0, (1.1) experiences Turing instability.Noticing that det(L i ) is a quadratic function about µ i , (3.14) and can confirm that (2.3) has at least one root with positive real part.Simple calculation suggests that is the ultimate condition.
Proof.Under the assumption in this theorem, conditions in Theorem 4.1 hold and (1.1) experiences the Turing instability.In addition, Corollary 3.4 guarantees the existence of nonconstant solutions in (1.2) provided that D 2 > D * .That is to say, the nonconstant solutions are generated by Turing instability and Turing patterns follow.

Steady state bifurcation
In this subsection and in next one, we assume that all eigenvalues µ i of −∆ are simple.Choose c as the bifurcation parameter and rewrite (2.4) into It is well known from [36] that the bifurcation point c S of steady state bifurcation satisfies Pattern formation of a Schnakenberg-type plant root hair initiation model 13 (H 2 ) there exists an i ∈ N 0 such that D i (c S ) = 0, T i (c S ) = 0, and D j (c S ) = 0, T j (c S ) = 0 for j = i; Indeed, D 0 (c) = r(k 2 u * 2 + k 1 ) > 0 for any c > 0, so we just check i ∈ N.
Next, we are devoted to finding c which satisfies (H 2 ).Let us define Then D i (c) = 0 is equivalent to D(c, p) = 0, that is, {(c, p) ∈ R 2 + : D(c, p) = 0} is the steady state bifurcation curve.Solving this equation demonstrates to be potential steady state bifurcation points.
In order to determine possible bifurcation points, we again solve D(c, p) = 0 and have with K > 0. To reach our goal, the following lemma is important.
This shows us that for any critical point p of c(p), c (p) > 0, therefore, the critical point must be unique and a local minimum point.On the other hand, it is easy to check lim p→0 + c(p) = lim p→+∞ c(p) = +∞, thus, the unique critical point p * is the global minimum point.Furthermore, because of the similarity between curves {(c(p), p)} and {(c, p ± (c))}, the properties about p ± (c) are obtained.

Hopf bifurcation
In this part, spatially homogeneous and nonhomogeneous periodic solutions of (1.1) are focused.Inspecting T i (c) and D i (c) in (4.4), a Hopf bifurcation point c H should meet (H 3 ) there exists i ∈ N 0 such that T i (c H ) = 0, D i (c H ) > 0, and T j (c H ) = 0, D j (c H ) = 0 for j = i; and the unique pair of complex eigenvalues near the imaginary axis α(c) ± iω(c) maintains Consequently, c H 0 is a Hopf bifurcation point for spatially homogeneous periodic solutions.
Hereafter, we intend to investigate spatially nonhomogeneous Hopf bifurcation for i ≥ 1.If c H i for some i ∈ N is a Hopf bifurcation point, then This gives that there is n ∈ N such that possible bifurcation points c Clearly, T i (c H i ) = 0 and T j (c H i ) = 0 for j = i.And another thing is to verify that D i (c H i ) > 0. Plugging (4.8) into D i (c) and still denoting µ i to p, we have 2 ] > 0, hence there should be two roots p − < 0 < p + of D i (c H i , p) combining with primary analysis.This produces that as long as 0 < µ i < p + we get D i (c H i ) > 0. Furthermore, D j (c H i ) = 0 if c H i = c S j for k + 1 ≤ j ≤ m.We then have, making use of simple calculation, According to previous discussion, we show the following Hopf bifurcation theorem.
Theorem 4.5.Suppose that Ω is a bounded smooth domain so that its spectral set S = {µ i : i ≥ 0} maintains (i) All eigenvalues µ i (i ≥ 0) are simple; (ii) There exists n ∈ N such that 0 < µ i < p + (1 ≤ i ≤ n), where p + is defined above.
(iii) c H i = c S j for any 1 ≤ i ≤ n and k + 1 ≤ j ≤ m, where c S j are defined in Theorem 4.4.
Then we find n + 1 Hopf bifurcation points c H i (0 ≤ i ≤ n) of (1.1) satisfying (4.9).In addition, 1.There is a smooth curve Ξ i of positive periodic orbits of (1.1) bifurcating from (c, u, v) = (c H i , u c H i , v c H i ), Ξ i contained in a global branch P i of positive periodic orbits of (1.1).
2. Occurrence of Hopf bifurcation at c = c H 0 also needs D 1 ≥ D 2 ; the bifurcating periodic orbits from c = c H 0 are spatially homogeneous.
3. The bifurcating periodic orbits from c = c H i (i = ) (j = i) or contains a spatially homogeneous periodic orbit on P 0 , or the projection Proj c P i contains the interval (0, c H i ) or (c H i , c * ), or there is c ∈ (0, c * − δ) such that for a sequence of periodic orbits (c s , u s , v s ) ∈ P i , c s → c and T s → ∞ as s → ∞, where T s is the period of (c s , u s , v s ).
Proof.Theorem 3.3 in [33] linking with general version of global bifurcation theorem similar to global steady state bifurcation in [27] give us the results about global Hopf bifurcation.Remark 4.6.Certainly, c H i = c S j for some 1 ≤ i ≤ n and k + 1 ≤ j ≤ m may lead Hopf-zero bifurcation, which is not in our scope.