The criteria for spreading and vanishing

Here we first give the comparison principle. The proof is similar to the proof of [7, Lemma 3.5].

Lemma 3.9. Let h¯ ∈ C¹([0,∞)), ¯u, ¯v ∈ C(D)×C^2,1(D), with D := {(x,t) ∈ R² : 0 < x <

h¯(t),t>0}. Assume that d_i >0,r_i >0,a_i >0,b_i >0and(u, ¯¯ v, ¯h)satisfies















¯

u_t−d₁u¯_xx+β₁u¯_x ≥u¯(r₁−a₁u¯), 0< x<h^¯(t),

¯

vt−d₂v¯xx+β₂v¯x ≥v¯(r₂−a₂v¯), 0< x<h^¯(t),

¯

u(_0,t)≥_{0, ¯}v(_0,t)≥_{0, ¯}u(h^¯(t)_,t) =_{0, ¯}v(h^¯(t)_,t) =_0, t>_0, h¯⁰(t)≥ −µ[u¯_x(h^¯(t),t) +ρv¯_x(h^¯(t),t)], t>0.

(3.12)

Assume that d_i >0,r_i >0,a_i >0,b₁> 0,b₂ <0and(u, ¯¯ v, ¯h)satisfies















¯

u_t−d₁u¯_xx+β₁u¯_x ≥u¯(r₁−a₁u¯), 0<x< h^¯(t),

¯

vt−d₂v¯xx+β₂v¯x≥ v¯(r₂−a₂v¯−b₂u¯), 0<x< h^¯(t),

¯

u(_0,t)≥_{0, ¯}v(_0,t)≥_{0, ¯}u(h^¯(t)_,t) =_{0, ¯}v(h^¯(t)_,t) =_0, t>_0, h¯⁰(t)≥ −µ[u¯_x(h^¯(t),t) +ρv¯_x(h^¯(t),t)], t>0.

(3.13)

Ifh¯(0) ≥ h₀, ¯u(x, 0) ≥ u₀(x)andv¯(x, 0) ≥ v₀(x) on [0,h₀], then the solution (u,v,h)of problem (1.5)satisfiesh¯(t)≥h(t)on[0,∞)andu¯ ≥u, ¯v≥v on[0,h(t)]×[0,∞).

Letλ⁽₁ⁱ⁾(l)be the principle eigenvalue of the following problem fori=1, 2 (−d_iφ_xx+β_iφ_x =λ⁽₁ⁱ⁾(l)φ, 0< x<l,

φ(0) =φ(l) =0. (3.14)

It is well known thatλ₁⁽ⁱ⁾(l) = _2d^βi

i

2

+d_i ^π_l2

is a strictly decreasing and continuous function inland

liml→0λ⁽₁ⁱ⁾(l) =_∞, lim

l→_∞λ⁽₁ⁱ⁾(l) = β_i

2d_i 2

.

Theorem 3.10. Suppose that d_i >0,r_i > 0,a_i > 0,b₁ > 0,b₂ ∈ _{R, 0}≤ β_i <2√

d_ir_i for i=1, 2. If h_∞ <_{∞, then h∞} ≤h^∗ =min{L_ri,i=1, 2}, where L_ri satisfiesλ₁⁽ⁱ⁾(L_ri) =r_i.

Proof. Due to Theorem3.6, limt→_∞ku(·,t),v(·,t)k_C1([0,h(t)]) = 0 ifh_∞ < _{∞. Assume} h_∞ > h^∗ to get a contradiction. Ifh_∞ > L_r1, then there exists ε >0 such thath_∞ > L_r1−b1ε. For suchε, there existsT₀1 such thath(T₀) =l> L_r1−b1ε andv(x,t)≤εfort ≥T₀, 0≤ x≤h(t).

Letz= z(x,t)be the unique solution of











z_t−d₁z_xx+β₁z_x =z(r₁−a₁z−b₁ε), 0<x <l,t ≥T₀, z(_0,t) =z(l,t) =_0, t ≥T₀,

z(x,T₀) =u(x,T₀), 0≤x ≤l.

Applying the comparison principle,z(x,t)≤ u(x,t) fort ≥ T₀, 0 ≤ x ≤ l. Since l > L_r1−b1ε, thenkz(·,t)−Z(·)k_C([0,l])→0 ast →_∞, whereZ(x)is the unique positive solution of

(−d₁Zxx+β₁Zx = Z(r₁−a₁Z−b₁ε), 0< x<l, Z(0) =Z(l) =0.

lim inft→_∞u(x,t)≥_limt→∞z(x,t) =Z(x)>_{0 in}(_0,l). This is a contradiction. Ifh_∞ >L_r2, we can get a contradiction by using the similar argument.

Lemma 3.11. Suppose that r_i >0,a_i > 0,b_i >0, 0≤ β_i <2√

d_ir_i,d_i =1for i=1, 2and h₀ <h^∗, then there existsµ>0depending on u0and v0such that h_∞ < _∞ifµ≤µ.

Proof. Sinceh₀ <h^∗, then λ⁽₁ⁱ⁾(h₀) = _2d^βi

i

2

+d_i ^2π_h

0

2

> r_i (i =1, 2). We can chooseδ,γsmall such that

1 h₀(1+δ)

d_iπ²

h₀(1+δ)−δγh₀

+ ^β

2i

4 −γ−r_i >0.

Define where M is a positive constant to be determined.

Direct computations yield

Lemma 3.12. Suppose that d_i >0,r_i >0,a_i > 0,b_i ∈_{R, 0}≤ β_i <2√

d_ir_i for i= 1, 2and h₀ <h^∗, there existsµ¯ >0such that h_∞ =_∞ifµ≥µ.¯

Proof. Due to the boundedness ofuandv, there existsδ^∗ such that

u(r₁−a₁u−b₁v)≥ −δ^∗u, v(r₂−a₂v−b₂u)≥ −δ^∗v.

Consider the following problem























wt−d₁wxx+β₁wx =−δ^∗w, 0< x<r(t),t>0, z_t−d₂z_xx+β₂z_x =−δ^∗z, 0< x<r(t)_,t>_0, w(0,t) =z(0,t) =0, t>0,

w(r(t),t) =z(r(t),t) =0, t>0, r⁰(t) =−µ[w_x(r(t),t) +ρz_x(r(t),t)], t>0, w(x, 0) =u₀(x),z(x, 0) =v₀(x),r(0) =h₀, 0< x<h₀.

(3.15)

Similar to Lemma 3.1, such problem admits a unique global solution (w,z,r). Applying the comparison principle, it follows that

u(x,t)≥w(x,t), v(x,t)≥ z(x,t), h(t)≥r(t), forx ∈[0,r(t)], t> 0. (3.16) Next, we prove that for all largeµ,r(₁)≥h^∗. Choose a smooth functionr(t)_{such that}

r(0) =h₀/2, r(1) =h^∗, r⁰(t)>0, fort>0.

Consider the following initial-boundary value problem



















w_t−d₁w_xx+β₁w_x =−δ^∗w, 0< x<r(t),t>0, z_t−d₂z_xx+β₂z_x =−δ^∗z, 0< x<r(t),t>0, w(0,t) =z(0,t) =0, t>0,

w(r(t),t) =z(r(t),t) =0, t>0,

w(x, 0) =w₀(x),z(x, 0) =z₀(x), 0< x<h₀/2,

(3.17)

here(w₀(x),z₀(x))satisfies

0<w₀(x)≤u₀(x) on [0,h₀/2], w₀(0) =w₀(h₀/2) =0, w⁰₀(h₀/2)<0, 0<z₀(x)≤v₀(x) on[0,h₀/2], z₀(0) =z₀(h₀/2) =0, z⁰₀(h₀/2)<0.

The standard theory for parabolic equations ensures that (3.17) has an unique positive solution (w,z) and w_x(r(t),t) < 0,z_x(r(t),t) < 0 for all t ∈ [0, 1] due to the Hopf Lemma.

Then there exists a constant ¯µ>0 such that for allµ≥µ,¯

r⁰(t)≤ −µ[w_x(r(t),t) +ρz_x(r(t),t)] fort∈[0, 1]. (3.18) Since the choice of initial values and (3.15)–(3.18), we have

r(t)≥r(t), w(x,t)≥w(x,t),z(x,t)≥z(x,t), for x∈[0,r(t)], t ∈[0, 1],

which implies r(1) ≥ r(1) = h^∗. In view of (3.16), h_∞ > h(1) ≥ h^∗. Together with Theo-rem3.10, derives the desired result.

Theorem 3.13. Assume that r_i >0,a_i >0,b_i >0, 0≤ β_i <2√

d_ir_i,d_i =1for i=1, 2and h₀ <h^∗, there exists µ^∗ ≥ µ∗ > 0 such that vanishing happens (h_∞ < _{∞) if} 0 < µ ≤ µ∗ or µ = µ^∗, and spreading happens (h_∞= _{∞) if}µ>µ^∗.

Proof. DefineΓ := {µ > 0 : h_∞ ≤ h^∗}. Due to Lemma3.11, Γ 6= ∅. In view of Lemma3.12, µ^∗ :=supΓ∈ [µ, ¯µ]. By the definition ofµ^∗ and Theorem 3.10, we get thath_∞= _∞ifµ>µ^∗.

Next, we prove that h_∞ < _{∞ if} µ = µ^∗. If not, h_∞ = _{∞ for} µ = µ^∗. So there exists T such thath(T)> h^∗. Since the solution(u,v,h)depends onµ, we write(uµ,vµ,hµ)instead of (u,v,h). By the continuous dependence of(u_µ,v_µ,h_µ)on µ, for smallε >0,h_µ(T)>h^∗for all [µ^∗−_ε,µ^∗+ε]_{. Then supΓ}≤ µ^∗−ε, which contradicts to the definition ofµ^∗. Henceµ^∗ ∈ _Γ.

DenoteΛ := {ν > 0 : ν ≥ µsuch thath_∞ ≤ h^∗ for all 0 < µ≤ ν}andµ∗ := supΛ≤ µ^∗. Using the similar way to the above, we obtain thatµ∗ ∈Λ. The proof is completed.

4 Acknowledgments

This work is supported by the National Natural Science Foundation of China (11771407), the Innovative Research Team of Science and Technology in Henan Province (17IRTSTHN007), and the National Key R&D Program of China (2017YFB0702504).

References

[1] A. A. Berryman, The orgins and evolution of predator–prey theory, Ecology 73(1992), No. 5, 1530–1535.https://doi.org/10.2307/1940005

[2] C. Bianca, Mathematical modeling for keloid formation triggered by virus: malignant effects and immune system competition,Math. Models Methods. Appl. Sci.21(2011), No. 2, 389–419.https://doi.org/10.1142/S021820251100509X;MR2776673

[3] C. Bianca, M. Pennisi, S. Motta, M.A. Ragusa, Immune system network and cancer vaccine, AIP Conference Proceedings 1389(2011), No. 1, 945–948. https://doi.org/10.

1063/1.3637764

[4] C. Bianca, F. Pappalardo, S. Motta, M. A. Ragusa, Persistence analysis in a Kolmogorov-type model for cancer-immune system competition,AIP Conference Proceed-ings 1558(2013), No. 1, 1797–1800.https://doi.org/10.1063/1.4825874

[5] R. S. Cantrell, C. Cosner, Spatial ecology via reaction–diffusion equations, Wiley Series in Mathematical and Computational Biology, John Wiley & Sons Ltd., Chichester, 2003.

https://doi.org/10.1002/0470871296;MR2191264

[6] J. Dockery, V. Hutson, K. Mischaikow, M. Pernarowski, The evolution of slow dis-persal rates: a reaction diffusion model, J. Math. Biol. 37(1998), No. 1, 61–83. https:

//doi.org/10.1007/s002850050120;MR1636644

[7] Y. Du, Z. Lin, Spreading–vanishing dichotomy in the diffusive logistic model with a free boundary,SIAM J. Math. Anal.42(2010), No. 1, 377–405.https://doi.org/10.1137/

090771089;MR2607347

[8] J. Guo, C. Wu, On a free boundary for a two-species weak competition system, J. Dyn. Differential Equations 24(2012), No. 4, 873–895. https://doi.org/10.1007/

s10884-012-9267-0;MR3000608

[9] X. He, W. Ni, Global dynamics of the Lotka–Volterra competition–diffusion system: dif-fusion and spatial heterogeneity I, Comm. Pure. Appl. Math. 69(2016), No. 5, 981–1014.

https://doi.org/10.1002/cpa.21596;MR3481286

[10] C. Lei, Z. Lin, H. Wang, The free bondary problem describing information diffusion in online social networks, J. Differential Equations 254(2013), No. 3, 1326–1341. https:

//doi.org/10.1016/j.jde.2012.10.021;MR2997373

[11] Y. Kaneko, Y. Yamada, A free boundary problem for a reaction diffusion equation ap-pearing in ecology,Adv. Math. Sci. Appl.21(2011), No. 2, 467–492.MR2953128

[12] Y. Lou, F. Lutscher, Evolution of dispersal in open advective environments, J. Math.

Biol. 69 (2014), No. 6–7, 1319–1342. https://doi.org/10.1007/s00285-013-0730-2;

MR3275198

[13] F. Lutscher, E. McCauley, M. A. Lewis, Spatial patterns and coexistence mechanisms in systems with unidirectional flow, Theor. Popul. Biol. 71(2007), No. 3, 267–277. https:

//doi.org/10.1016/j.tpb.2006.11.006

[14] Y. Kuang, E. Beretta, Global qualitative analysis of a ratio-dependent predator–prey sys-tem, J. Math. Biol.36(1998), No. 4, 389–406. https://doi.org/10.1007/s002850050105;

MR1624192

[15] K. Y. Lam, W. Ni, Uniqueness and complete dynamics in heterogeneous competition diffusion systems,SIAM J. Appl. Math.72(2012), No. 6, 1695–1712.https://doi.org/10.

1137/120869481;MR3022283

[16] Y. Lou, D. Xiao, P. Zhou, Qualitative analasis for a Lotka–Volterra competition system in advective homogeneous environment, Discrete Contin. Dyn. Syst. A. 36(2016), No. 2, 953–969.https://doi.org/10.3934/dcds.2016.36.953;MR3392913

[17] Y. Lou, P. Zhou, Evolution of dispersal in advective homogeneous environment: the effect of boundary conditions, J. Differential Equations 259(2015), No. 1, 141–171. https:

//doi.org/10.1016/j.jde.2015.02.004;MR3335923

[18] M. H. Protter, H. F. Weinberger, Maximum principles in differential equations, Springer-Verlag, New York, 1984.https://doi.org/10.1007/978-1-4612-5282-5;MR0762825 [19] C. V. Pao, Nonlinear parabolic and elliptic equations, Springer Science & Business Media,

New York, 1992.MR1212084

[20] J. Ren, X. Li, Bifurcations in a seasonally forced predator–prey model with generalized Holling type IV functional response, Int. J. Bifurcation Chaos 26(2016), No. 12, 1650203.

https://doi.org/10.1142/S0218127416502035;MR3574812

[21] J. Ren, X. Li, How seasonal forcing influences the complexity of a predator–prey system, Discrete Contin. Dyn. Syst. B.23(2018), No. 2, 785–807.https://doi.org/10.3934/dcdsb.

2018043

[22] J. Ren, L. Yu, Codimension-two bifurcation, chaos and control in a discrete-time informa-tion diffusion model,J. Nonlinear Sci. 26(2016), No. 6, 1895–1931. https://doi.org/10.

1007/s00332-016-9323-8;MR3562398

[23] H. L. Smith, Monotone dynamical systems. An introduction to the theory of competitive and cooperative systems,Mathematical Surveys and Monographs, Vol. 41 American Mathemat-ical Society, Providence, 1995.MR1319817

[24] M. Wang, On some free boundary problems of the prey-predator model, J. Differential Equations256(2014), No. 10, 3365–3394. https://doi.org/10.1016/j.jde.2014.02.013;

MR3177899

[25] F. Wang, H. Wang, K. Xu, Diffusive logistic model towards predicting information diffu-sion in online social networks, in: 32nd International Conference on Distributed Computing Systems Workshops (ICDCS Workshops)2012, 133–139.https://doi.org/10.1109/ICDCSW.

2012.16

[26] M. Wang, J. Zhao, Free boundary problems for a Lotka–Volterra competition sys-tem, J. Dyn. Differential Equations 26(2014), No. 3, 655–672. https://doi.org/10.1007/

s10884-014-9363-4;MR3274436

[27] M. Wang, Y. Zhang, Two kinds of free boundary problems for the diffusive prey–

predator model,Nonlinear Anal. Real World Appl. 24(2015), 73–82.https://doi.org/10.

1016/j.nonrwa.2015.01.004;MR3332883

[28] X. Zhao,Dynamical systems in population biology, Springer, New York, 2003.MR1980821 [29] P. Zhou, On a Lotka–Volterra competition system: diffusion vs advection, Calc. Var.

55(2016), No. 6.https://doi.org/10.1007/s00526-016-1082-8;MR3566937

[30] L. Zhou, S. Zhang, Z. Liu, A free boundary problem of a predator–prey model with advection in heterogeneous environment, Appl. Math. Comput. 289(2016), 22–36. https:

//doi.org/10.1016/j.amc.2016.05.008;MR3515835

In document fixed boundary or free boundary (Pldal 25-31)