i

i

“main” — 2015/12/7 — 21:59 — page 1 — #1

i

i

Bull Braz Math Soc, New Series 47(1), 1-14 © 2016, Sociedade Brasileira de Matemática ISSN: 1678-7544 (Print) / 1678-7714 (Online)

A mixed hyperbolic-parabolic system to describe predator-prey dynamics Elena Rossi Abstract. Following [2], a model aiming at the description of two competing populations is introduced. In particular, it is considered a nonlinear system consisting of a nonlocal conservation law for predators coupled with a parabolic equation for prey. The drift term in the equation for predators is in general a nonlocal and nonlinear function of the prey density: the movement of predators can hence be directed towards regions where the concentration of prey is higher. Lotka-Volterra type right hand sides describe the feeding. In [2] the resulting Cauchy problem is proved to be well posed in any space dimension with respect to the L1 topology, and estimates on the growth of the solution in L1 and L∞ norm and on the time dependence are provided. Numerical integrations show a few qualitative features of the solutions. This is a joint work with Rinaldo M. Colombo. Keywords: nonlocal conservation laws, predatory-prey systems, mixed hyperbolicparabolic problems. Mathematical subject classification: 35L65, 35M30, 92D25.

1 Introduction Consider the predator-prey model introduced in [2]  ∂t u + div (u v(w)) = (α w − β) u ∂t w − μ w = (γ − δ u) w

(1.1)

where u = u(t, x ) and w = w(t, x ) are the density at time t ∈ R+ and position x ∈ Rn of predators and prey respectively. The parameters α, β, γ , δ appearing Received 11 March 2015. The author was supported by the PRIN 2012 project Nonlinear Hyperbolic Partial Differential Equations, Dispersive and Transport Equations: Theoretical and Applicative Aspects and by the GNAMPA 2014 project Conservation Laws in the Modeling of Collective Phenomena.

i

i i

i

i

i

“main” — 2015/12/7 — 21:59 — page 2 — #2

i

2

i

ELENA ROSSI

in system (1.1) are all positive, μ is strictly positive. (1.1) is a mixed system consisting of a balance law and a parabolic equation. The former describes the evolution of predators density u, the latter the evolution of prey density w. The source terms are motivated by Lotka-Volterra equations. The coefficient α in the source term of the balance law accounts for the increase in the predators density due to feeding on prey, while β is the predators mortality rate. Concerning the parabolic equation, γ is the prey birth rate and δ the prey mortality rate due to predators. The basic model by Lotka and Volterra, based on ordinary differential equations, implicitly assumes a homogeneous distribution of the two populations in space. The advantage of model (1.1) is to allow also for spacial variations of predators and prey. More precisely, prey w are supposed to diffuse in the whole space without preferred direction, while the movement of predators u can be directed towards the regions where there is a higher concentration of prey. Indeed, the flow u v(w) accounts for the direction preferred by predators. The velocity v is in general a nonlocal and nonlinear function of the prey density. A typical choice can be ∇(w ∗ η) , (1.2) v(w) = κ  1 + ∇(w ∗ η)2 meaning that predators move towards regions ofhigher concentrations of prey. Here, when η is a positive smooth mollifier with Rn η dx = 1, the space convolution product (w(t) ∗ η) (x ) has the meaning of an average of the prey density at time t around position x . The denominator in (1.2) is a smooth normalisation factor, so that the positive parameter κ is the maximal predators speed. A key feature of model (1.1) is that predators may have a well defined horizon which defines how far they can “feel” the presence of prey. This horizon is represented, for example, by the radius of the support of η in (1.2). In [2] the class of models (1.1) has been studied under suitable assumptions on v. Existence, uniqueness, continuous dependence from the initial datum and various stability estimates for the solutions to (1.1) have been proved. In particular, solutions are found in the space L1 ∩ L∞ ∩ BV for the predators and in L1 ∩ L∞ for the prey, and are thus understood in the distributional sense, see Definitions 2.1, 2.3 and 2.5. Moreover, all analytic results hold in any space dimension. In [10], an explicit numerical scheme for the discretisation of the system (1.1) in two space dimensions is introduced and its convergence is proved. The paper is organised as follows. The next Section presents the main analytical results, while Section 3 is devoted to sample numerical integrations of (1.1) in two space dimensions, using the algorithm introduced in [10]. Bull Braz Math Soc, Vol. 47, N. 1, 2016

i

i i

i

i

i

“main” — 2015/12/7 — 21:59 — page 3 — #3

i

i

HYPERBOLIC-PARABOLIC SYSTEM TO DESCRIBE PREDATOR-PREY DYNAMICS

3

2 Analytical results This section is devoted to the main analytical result, the well posedness of system (1.1), and the propositions at its basis. For the proofs we refer to [2]. We first give a rigorous definition of solution to (1.1). Definition 2.1. Let T > 0 be fixed. A solution to the system (1.1) on [0, T ] is a pair (u, w) ∈ C0 ([0, T ]; L1 (Rn ; R2 )) such that • setting a(t, x ) = γ − δ u(t, x ), w is a weak solution to ∂t w − μ w = a w; • setting b(t, x ) = α w(t, x ) − β and c(t, x ) = v(w(t))(x ), u is a weak solution to ∂t u + div (u c) = b u. The extension to the case of the Cauchy problem is immediate. Throughout, we work in the spaces     X = (L1 ∩ L∞ ∩ BV) Rn ; R × (L1 ∩ L∞ ) Rn ; R and X + = (L1 ∩ L∞ ∩ BV)(Rn ; R+ ) × (L1 ∩ L∞ )(Rn ; R+ )

with the norm (u, w)X = uL1 (Rn ;R) + wL1 (Rn ;R) .

(2.1)

System (1.1) is defined by a few real positive parameters and by the map v, which is assumed to satisfy the following condition: (v) v : (L1 ∩ L∞ )(Rn ; R) → (C2 ∩ W1,∞ )(Rn ; Rn ) admits a constant K and ∞ (R+ ; R+ ) such that, for all w, w1 , w2 ∈ (L1 ∩ an increasing map C ∈ Lloc ∞ n L )(R ; R), v(w)L∞ (Rn ;Rn ) ≤ K wL1 (Rn ;R) ∇v(w)L∞ (Rn ;Rn×n ) ≤ K wL∞ (Rn ;R) v(w1 ) − v(w2 )L∞ (Rn ;Rn ) ≤ K w1 − w2 L1 (Rn ;R)   ∇ (div v(w))L1 (Rn ;Rn ) ≤ C wL1 (Rn ;R) wL1 (Rn ;R)   div (v(w1 ) − v(w2 ))L1 (Rn ;R) ≤ C w2 L∞ (Rn ;R) w1 − w2 L1 (Rn ;R) . The bound on the L∞ norm of v(w) by means of the L1 norm of w, see the first and the third inequality above, is typical of a nonlocal, e.g. convolution, operator. Under reasonable regularity conditions on the kernel η, Lemma 4.1 in [2] ensures that the operator v in (1.2) satisfies (v). Bull Braz Math Soc, Vol. 47, N. 1, 2016

i

i i

i

i

i

“main” — 2015/12/7 — 21:59 — page 4 — #4

i

4

i

ELENA ROSSI

Relying solely on the assumption (v), we state the main analytical result. Theorem 2.2. Fix α, β, γ , δ ≥ 0 and μ > 0. Assume that v satisfies (v). Then, there exists a map R : R+ × X + → X + with the following properties: 1. R is a semigroup: R0 = Id and Rt2 ◦ Rt1 = Rt1 +t2 for all t1 , t2 ∈ R+ . 2. R solves (1.1): for all (u o , wo ) ∈ X + , the map t → Rt (u o , wo ) solves the Cauchy Problem ⎧ ∂t u + div (u v(w)) = (α w − β) u ⎪ ⎪ ⎨ ∂t w − μ w = (γ − δ u) w u(0, x ) = u o (x ) ⎪ ⎪ ⎩ w(0, x ) = wo (x ) in the sense of Definition 2.1. In particular, for all (u o , wo ) ∈ X + the map t → Rt (u o , wo ) is continuous in time. 3. Local Lipschitz continuity in the initial datum: for all r > 0 and for all t ∈ R+ , there exist a positive L(t, r) such that for all (u 1, w1 ), (u 2 , w2 ) ∈ X + with u i L∞ (Rn ;R) ≤ r,

TV (u i ) ≤ r,

wi L∞ (Rn ;R) ≤ r,

wi L1 (Rn ;R) ≤ r

for i = 1, 2, the following estimate holds: Rt (u 1 , w1) − Rt (u 2 , w2 )X ≤ L(t, r) (u 1 , w1 ) − (u 2 , w2 )X . 4. Growth estimates: for all (u o , wo ) ∈ X + and for all t ∈ R+ , denote (u, w)(t) = Rt (u o , wo ). Then,  γt e −1 u(t)L1 (Rn ;R) ≤ u o L1 (Rn ;R) exp α wo L∞ (Rn ;R) γ  eγ t − 1 u(t)L∞ (Rn ;R) ≤ u o L∞ (Rn ;R) exp (α + K ) wo L∞ (Rn ;R) γ w(t)L1 (Rn ;R) ≤ wo L1 (Rn ;R) eγ t w(t)L∞ (Rn ;R) ≤ wo L∞ (Rn ;R) eγ t . 5. Propagation speed: if (u o , wo ) ∈ X + is such that spt(u o ) ⊆ B(0, ρo ), then for all t ∈ R+ , spt (u(t)) ⊆ B (0, ρ(t))

where

ρ(t) = ρo + K t eγ t wo L1 (Rn ;R) .

Bull Braz Math Soc, Vol. 47, N. 1, 2016

i

i i

i

i

i

“main” — 2015/12/7 — 21:59 — page 5 — #5

i

HYPERBOLIC-PARABOLIC SYSTEM TO DESCRIBE PREDATOR-PREY DYNAMICS

i

5

An explicit estimate of the Lipschitz constant L(t, r) is provided in [2], Equation (4.37). The growth estimates in point 4 provide an a priori control on the total mass and on the peaks of the solution. Point 5 means that the solution to the balance law propagates with finite speed. In [2], Theorem 2.2 is proved thanks to a fixed point argument and to careful estimates on the parabolic problem  ∂t w − μ w = a(t, x ) w (2.2) w(to , x ) = wo (x ) and, separately, on the balance law  ∂t u + div (c(t, x ) u) = b(t, x ) u u(to , x ) = u o (x )

(2.3)

with suitable assumptions on the functions a, b and c. The approaches to both the evolution equations (2.2) and (2.3) are identical. We recall below the key definitions and the basic well posedness results. Throughout, we fix to , T ∈ R+ , with T > to , and denote 

(2.4) I = [to , T ] and J = (t1 , t2) ∈ I 2 : t1 < t2 . To improve the readability of the statements below, we denote by O (t) an increasing smooth function of time t, depending on the space dimension n and on various norm of the coefficients a, μ in (2.2) and b, c in (2.3). We recall the following notions from the theory of parabolic equations. They are similar to various results in the wide literature on parabolic problems, see for instance [1, 8, 9], but here we are dealing with L1 solutions on the whole space. Inspired by [9, Section 48.3], we give the following definition, where we used the notation (2.4). Definition 2.3. Let a ∈ L∞ (I × Rn ; R) and wo ∈ L1 (Rn ; R). A weak solution to (2.2)is a function w ∈ C0 (I ; L1 (Rn ; R)) such that for all test functions ϕ ∈ C1 I ; C2c (Rn ; R)  T (w ∂t ϕ + μ w ϕ + a w ϕ) dx dt = 0 (2.5) to

Rn

and w(to , x ) = wo (x ). The following Proposition concerns the well posedness of (2.2) and provide some useful stability estimates. Proposition 2.4. Let a ∈ L∞ (I × Rn ; R). Then, the Cauchy problem (2.2) generates a map P : J × L1 (Rn ; R) → L1 (Rn ; R) with the following properties: Bull Braz Math Soc, Vol. 47, N. 1, 2016

i

i i

i

i

i

“main” — 2015/12/7 — 21:59 — page 6 — #6

i

6

i

ELENA ROSSI

1. P is a Process. 2. P solves (2.2) in the sense of Definition 2.3 for all wo ∈ L1 (Rn ; R). 3. Regularity in wo: for all (to , t) ∈ J , the map Pto ,t : L1 (Rn ; R) → L1 (Rn ; R) is linear and continuous, with   Pt ,t wo  1 n ≤ O (t) wo L1 (Rn ;R) , o L (R ;R) and, moreover, for all wo ∈ (L1 ∩ L∞ )(Rn ; R), for all (to , t) ∈ J ,   Pt ,t wo  ∞ n ≤ O (t) wo L∞ (Rn ;R) . o L (R ;R) 4. Stability in a: let a1 , a2 ∈ L∞ (I × Rn ; R) with a1 − a2 ∈ L1 (I × Rn ; R) and call P 1 , P 2 the corresponding processes. Then, for all (to , t) ∈ J and for all wo ∈ (L1 ∩ L∞ )(Rn ; R),   1 P wo − P 2 wo  1 n ≤ O (t) wo L∞ (Rn ;R) a1 − a2 L1 ([t ,t ]×Rn ;R) . to ,t to ,t o L (R ;R) 5. Positivity: if wo ∈ (L1 ∩ L∞ )(Rn ; R) and wo ≥ 0, then Pto ,t wo ≥ 0 for all (to , t) ∈ J . 6. Regularity in (t, x): if wo ∈ (L1 ∩C1 )(Rn ; R), then (t, x ) → (Pto ,t wo )(x ) ∈ C1(I × Rn ; R). For a detailed proof and the precise estimates we refer to [2, Proposition 2.5]. Although the literature on the equation (2.2) is vast, considering it in L1 ∩ L∞ on all Rn seems to be slightly unconventional. In [2] detailed proofs and estimates are provided, inspired by the results in [1, 4]. Consider now the hyperbolic problem (2.3), following the same template as for the parabolic equation. We first give the definition of weak solution to (2.3) in a way similar to [3, Section 4.3] and [11, Section 3.5]. We keep on using the notation (2.4). Definition 2.5. Let b ∈ L∞ (I × Rn ; R), c ∈ L∞ (I × Rn ; Rn ) and u o ∈ (L1 ∩ L∞ )(Rn ; R). A weak solution to (2.3) is a function u ∈ C0 (I ; L1 (Rn ; R)) such that for all test functions ϕ ∈ C1c ( I˚ × Rn ; R)  T (u ∂t ϕ + u c · ∇ϕ + b u ϕ) dx dt = 0 (2.6) to

Rn

and u(to , x ) = u o (x ). The following Proposition is analogous to Proposition 2.4. Proposition 2.6. Under the assumptions: Bull Braz Math Soc, Vol. 47, N. 1, 2016

i

i i

i

i

i

“main” — 2015/12/7 — 21:59 — page 7 — #7

i

HYPERBOLIC-PARABOLIC SYSTEM TO DESCRIBE PREDATOR-PREY DYNAMICS

i

7

(b) b ∈ (C1 ∩ L∞ )(I × Rn ; R); ∇b ∈ L1 (I × Rn ; Rn ); (c) c ∈ (C2 ∩ L∞ )(I × Rn ; Rn ); ∇c ∈ L∞ (I × Rn ; Rn×n ); ∇(div c) ∈ L1 (I × Rn ; Rn ), the Cauchy Problem (2.3) generates a map H : J × (L1 ∩ L∞ ∩ BV)(Rn ; R) → (L1 ∩ L∞ ∩ BV)(Rn ; R) with the following properties: 1. H is a process. 2. H solves (2.3) in the sense of Definition 2.5 for all u o ∈ (L1 ∩ L∞ ∩ BV)(Rn ; R). 3. Regularity in uo: for all (to, t) ∈ J the map Hto,t : (L1 ∩L∞ ∩BV)(Rn ; R) → (L1 ∩ L∞ ∩ BV)(Rn ; R) is linear and continuous, with   Ht ,t u o  1 n ≤ O (t) u o L1 (Rn ;R) , o L (R ;R) and, moreover, for all u o ∈ (L1 ∩ L∞ ∩ BV)(Rn ; R), for all (to , t) ∈ J ,   Ht ,t u o  ∞ n ≤ O (t) u o L∞ (Rn ;R) . o L (R ;R) 4. Stability in b, c: if b1, b2 satisfy (b) with b1 − b2 ∈ L1 (I × Rn ; R) and c1 , c2 satisfy (c) with div (c1 −c2 ) ∈ L1 (I ×Rn ; R), call H 1, H 2 the corresponding processes. Then, for all (to , t) ∈ J and for all u o ∈ (L1 ∩ L∞ ∩ BV)(Rn ; R),    1  Hto ,t u o − Ht2o ,t u o 

L1 (Rn ;R)

  ≤ O (t ) u o L∞ (Rn ;R) + TV (u o ) c1 − c2 L1 ([to ,t ];L∞ (Rn ;Rn ))   + O (t ) u o L∞ (Rn ;R) b1 − b2 L1 ([to ,t ]×Rn ;R) + ∇ · (c1 − c2 )L1 ([to ,t ]×Rn ;R) .

5. Positivity: if u o ∈ (L1 ∩ L∞ ∩ BV)(Rn ; R) and u o ≥ 0, then Hto ,t u o ≥ 0 for all (to , t) ∈ J . 6. Total variation bound: if u o ∈ (L1 ∩ L∞ ∩ BV)(Rn ; R), then, for all (to , t) ∈ J,     TV Hto ,t u o ≤ O (t) TV (u o ) + u o L∞ (Rn ;R) . for all uo ∈ (L1 ∩ L∞ ∩ BV)(Rn ; R), the 7. L1 Lipschitz continuity in time:  0,1 1 map t → Hto ,t u o is in C I ; L (Rn ; R) . 8. Finite propagation speed: let (to, t) ∈ J and u o ∈ (L1 ∩ L∞ ∩ BV)(Rn ; R) have compact support spt u o . Then, also spt Hto ,t u o is compact. Bull Braz Math Soc, Vol. 47, N. 1, 2016

i

i i

i

i

i

“main” — 2015/12/7 — 21:59 — page 8 — #8

i

8

i

ELENA ROSSI

For a detailed proof and the precise estimates we refer to [2, Proposition 2.8]. The proof exploits the classical results by Kružkov [5], while the assumptions (b) and (c) are needed to fit in the framework of [6, 7]. It is interesting to note that, even if both Cauchy Problems (2.2) and (2.3) are non autonomous problems and generate a process, the mixed system (1.1) generates a semigroup. Remark 2.7. Consider the following generalisation of (1.1):  ∂t u + div ( f (u) v(w)) = (α w − β) u ∂t w − μ w = (γ − δ u) w

(2.7)

∞ (R; R). Exiswith the assumptions f ∈ C2 (R; R), f (0) = 0 and ∂u f ∈ Lloc tence, uniqueness and stability estimates follow by a straightforward extension of Proposition 2.6 and Theorem 2.2.

3 Numerical Integrations An explicit numerical scheme for the discretisation of system (2.7), generalisation of (1.1), in two space dimensions has been introduced in [10], where also the convergence of the algorithm has been proved. To approximate (2.7), we use a finite-difference scheme for the parabolic part, while the balance law is integrated by means of a Lax-Friedrichs type finite volume method with dimensional splitting. Both the source terms are solved using a second order Runge-Kutta method, which guarantees the positivity of the approximate solution. As already said, we focus on the two-dimensional case and use the vector field v in (1.2), where the kernel function η is chosen as follows 

η(x) = ηˆ − x 2

2

3

χ B(0, ) (x) with ηˆ ∈ R

+



such that

R2

η(x) dx = 1 . (3.1)

We present the result of two numerical integrations to illustrate some qualitative properties of the solutions to (2.7). In particular, in both cases, we consider f (u) = u, i.e. we compute the solution to system (1.1). Below, we constrain both unknown functions u and w to remain equal to the initial datum all along the boundary of the computational domain. Concerning the balance law, this can be explained as assuming to have a constant value outside the computational domain and compute the flux accordingly. For the second equation, the choice of these boundary conditions amounts to assume that the displayed solution is part of a solution defined on all R2 that gives a constant inflow into the computational domain. Bull Braz Math Soc, Vol. 47, N. 1, 2016

i

i i

i

i

i

“main” — 2015/12/7 — 21:59 — page 9 — #9

i

i

HYPERBOLIC-PARABOLIC SYSTEM TO DESCRIBE PREDATOR-PREY DYNAMICS

9

3.1 Predators chasing preys This example present a situation in which the effect of the first order transport term in the equation for predators u can be easily seen. Moreover, it is clearly visible also the typical Lotka-Volterra effect, in which a species almost disappear and then its mass rises again. We set v as in (1.2), η as in (3.1) and α = 2 γ = 1

β = 1 δ = 2

κ = 1 μ = 0.5

= 0.0375

(3.2)

with the following initial datum on the numerical domain [0, 0.5] × [0, 1]:

u o (x , y) = 4 χ (x , y) A

wo (x , y) = 3 (2y − 1) max{0, h(x , y)} χ (x , y) B

(3.3)

where h(x , y) = (4x − 1)2 + (4y − 2)2 − 0.25 A = {(x , y) ∈ R2 : (2(4x − 1))2 + (1.25 (4y − 1))2 ≤ 1} B = {(x , y) ∈ R2 : y ≥ 0.5} . The solution is computed up to time Tmax = 0.3 and the result of the numerical integration is displayed in Figure 1. Bull Braz Math Soc, Vol. 47, N. 1, 2016

i

i i

i

i

i

“main” — 2015/12/7 — 21:59 — page 10 — #10

i

10

i

ELENA ROSSI

Figure 1: Numerical integration of (1.1)-(3.1)-(3.2) with initial datum (3.3). In each couple of figures, the predator density u is on the left and the prey density w is on the right. The darker the colour, the higher the density. First, predators decrease due to lack of nutrients, prey. Prey diffuse and reach the zone where they are “seen” by predators. Then, predators are attracted to prey: the former now increase, while the latter decrease. This solution was obtained with space mesh size x = y = 0.00125. Bull Braz Math Soc, Vol. 47, N. 1, 2016

i

i i

i

i

i

“main” — 2015/12/7 — 21:59 — page 11 — #11

i

HYPERBOLIC-PARABOLIC SYSTEM TO DESCRIBE PREDATOR-PREY DYNAMICS

i

11

At the beginning, predators can not see prey. Hence the former decrease, while the latter are free to increase, thanks to their birth rate. At the same time, prey diffuse, and some of them enter the region where predators feel their presence. This causes predators to move towards the area of highest prey density. Then, predators increase and their effect on the prey population is clearly seen, as shown also by the graph of the integrals of u and w in Figure 2.

Figure 2: The integrals of u, left, and w, right, over the computational domain versus time. (u, w) is the numerical solution to (1.1)-(3.1)-(3.2). We remark that, in the present setting, as time grows, undesired effects due to the presence of the boundary become relevant. 3.2 A Dynamic Equilibrium In this second example, the numerical solution displays an interesting asymptotic state. The diffusion caused by the Laplacian in the prey equation counterbalances the first order nonlocal differential operator in the predator equation: the outcome is the onset of a discrete, quite regular, structure, see Figure 3. We set v as in (1.2), η as in (3.1) and α = 1 γ = 0.4

β = 0.2 δ = 24

κ = 1 μ = 0.5

= 0.0625

(3.4)

with the following initial datum on the numerical domain [0, 0.5] × [0, 1] Bull Braz Math Soc, Vol. 47, N. 1, 2016

i

i i

i

i

i

“main” — 2015/12/7 — 21:59 — page 12 — #12

i

12

i

ELENA ROSSI

u o (x , y) = 0.25 χ (x , y) + 0.2 χ (x , y) C

D

wo (x , y) = 0.2

(3.5)

where C ={(x , y) ∈ R2 : (4x − 0.6)2 + (4y − 3)2 < 0.01} D ={(x , y) ∈ R2 : (4x − 1.3)2 + (4y − 0.8)2 < 0.04} . The solution is computed up to time Tmax = 3 on a mesh of width x = y = 0.00125 and the result is displayed in Figure 3. At first, predators almost disappear, while moving towards the central part of the numerical domain. Then, they start to increase. Slowly, a regular pattern arises. Predators focus in small regions regularly distributed, which display a fairly stable behaviour while passing from being arranged along 5 to along 6 columns, see Figure 3, second line. This pattern can be explained analytically as a dynamic equilibrium between the first order non local operator in the predator equation and the Laplacian in the prey equation. In the small regions where predators are concentrated, their feeding on prey causes “holes” in the prey density. As a consequence, by symmetry considerations, the average gradient of the prey density, which directs the movement of predators, almost vanishes and, hence, predators almost do not move. At the same time, the diffusion of prey keeps filling the “holes”, providing new nutrient to predators. Bull Braz Math Soc, Vol. 47, N. 1, 2016

i

i i

i

i

i

“main” — 2015/12/7 — 21:59 — page 13 — #13

i

HYPERBOLIC-PARABOLIC SYSTEM TO DESCRIBE PREDATOR-PREY DYNAMICS

i

13

Figure 3: Numerical integration of (1.1)-(3.1)-(3.4) with initial datum (3.5). In each couple of figures, the predator density u is on the left and the prey density w is on the right. The darker the colour, the higher the density. First, predators decrease and move towards the central region. Then, a discrete pattern arises: predators focus in small regions regularly distributed initially along 5 columns, later along 6 columns. This solution was obtained with space mesh size x = y = 0.00125. Bull Braz Math Soc, Vol. 47, N. 1, 2016

i

i i

i

i

i

“main” — 2015/12/7 — 21:59 — page 14 — #14

i

14

i

ELENA ROSSI

References [1]

[2] [3]

[4] [5] [6]

H. Amann. Linear and quasilinear parabolic problems. Vol. I, volume 89 of “Monographs in Mathematics”. Birkhäuser Boston Inc., Boston, MA, 1995. Abstract linear theory. R.M. Colombo and E. Rossi. Hyperbolic predators vs. parabolic prey. Commun. Math. Sci., 13(2) (2015), 369–400. C.M. Dafermos. Hyperbolic conservation laws in continuum physics, volume 325 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, third edition, (2010). A. Friedman. Partial differential equations of parabolic type. Prentice-Hall Inc., Englewood Cliffs, N.J., (1964). S.N. Kružkov. First order quasilinear equations with several independent variables. Mat. Sb. (N.S.), 81(123) (1970), 228–255. M. Lécureux-Mercier. Improved stability estimates for general scalar conservation laws. J. Hyperbolic Differ. Equ., 8(4) (2011), 727–757.

[7]

M. Lécureux-Mercier. Improved stability estimates on general scalar balance laws. ArXiv e-prints, July (2013).

[8]

A. Lunardi. Analytic semigroups and optimal regularity in parabolic problems. Progress in Nonlinear Differential Equations and their Applications, 16. Birkhäuser Verlag, Basel, (1995).

[9]

P. Quittner and P. Souplet. Superlinear parabolic problems. Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser Verlag, Basel, 2007. Blow-up, global existence and steady states.

[10] E. Rossi and V. Schleper. Convergence of a numerical scheme for a mixed hyperbolic-parabolic system in two space dimensions. ESAIM:M2AN, 2015. To appear. [11] D. Serre. Systems of conservation laws. 1&2. Cambridge University Press, Cambridge, 1999. Translated from the 1996 French original by I.N. Sneddon.

Elena Rossi Department of Mathematics and Applications University of Milano-Bicocca Via R. Cozzi, 55 20125 Milano ITALY E-mail: [email protected]

Bull Braz Math Soc, Vol. 47, N. 1, 2016

i

i i

i