Nonlinear Stability and Numerical Simulations for a Reaction–Diffusion

IJNSNS 2021; aop

Florinda Capone*, Maria Francesca Carfora, Roberta De Luca and Isabella Torcicollo Nonlinear stability and numerical simulations for a reaction–diffusion system modelling Allee effect on predators https://doi.org/10.1515/ijnsns-2020-0015 Received January 17, 2020; accepted May 31, 2021; published online August 5, 2021

Abstract: A reaction–diffusion system governing the prey–predator interaction with Allee effect onthe predators, already introduced by the authors in a previous work is reconsidered with the aim of showing destabilization mechanisms of the biologically meaning equilibrium and detecting some aspects for the eventual oscillatory pattern formation. Extensive numerical simulations, depicting such complex dynamics, are shown. In order to complete the stability analysis of the coexistence equilibrium, a nonlinear stability result is shown. Keywords: Alleeeffect;nonlinearstability;predator–prey;reaction–diffusion;Routh–Hurwitz;Turing–Hopf instabilities.

1 Introduction

Ecological systems are characterized by the interaction between species and their natural environment. An important type of interaction which influences dynamics of all species is predation. Thus predator–prey models have been in the focus of ecological science since the early days of this discipline. It has turned out that predator–prey systems can show different dynamical behaviors (steady states, oscillations, chaos) depending on the value of model parameters. In predator–prey interactions, a crucial role is played by the cooperation which can help the survival of individuals. In fact, hunting cooperation, observed in many species (carnivores, birds, aquatic organisms, and spiders), implies, as a direct consequence, Allee effect in predators which allows these last to persist even if the prey population does not sustain them in the absence of hunting cooperation [1]. In many papers addressing population dynamics it is assumed that population is homogeneously distributed in the environment, leading to the formulation of ordinary differential equations (ODEs) systems [1–6]. If spatial domains are considered a spatial predator–prey system can be considered as a reaction–diffusion system. Reaction–diffusion systems have been studied in many fields as physics,

*Corresponding author: Florinda Capone, Department of Mathematics and Applications ‘‘R.Caccioppoli’’, University of Naples Federico II, Via Cintia, Naples, Italy, E-mail: [email protected] Roberta De Luca, Department of Mathematics and Applications ‘‘R. Caccioppoli’’, University of Naples Federico II, Via Cintia, Naples, Italy, E-mail: [email protected] Maria Francesca Carfora and Isabella Torcicollo, Istituto per le Applicazioni del Calcolo ‘‘Mauro Picone’’, Via P. Castellino 111, CNR, Naples, Italy, E-mail: [email protected] (M.F. Carfora), [email protected] (I. Torcicollo)

Open Access. © 2021 Florinda Capone et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 International License. 2 | F. Capone et al.: On a reaction–diffusion system modelling Allee effect on predators chemistry, biology, economy and from many points of view ([7–13]). In particular it has been shown that reaction–diffusion systems are capable of self organized pattern formation. These spatial patterns arise not from inhomogeneity of initial or boundary conditions, but purely from the dynamics of the system, i.e. from the interaction of nonlinear reactions of growth processes and diffusion (as shown already by Turing14 [ ]). In particular the formation of stationary spatial patterns is due to Turing instabilities. The simultaneous appearance of Turing instability, which leads to steady spatial structures, with Hopf instability, which gives rise to temporal oscillations, is of great interest because these bifurcations are responsible for the breaking of spatial and temporal symmetries, respectively. The focus of research of many authors has been shifted to the study of the formation of such a type of spatio-temporal patterns. It has been shown that spatio-temporal patterns are very likely to be found in the neighborhood of Turing–Hopf bifurcations [15–17]. In this kind of bifurcation the formation of inhomogeneous stationary patterns caused by Turing instabilities “interacts” with the appearance of oscillations due to a Hopf bifurcation. In [7], the classical Lotka–Volterra model with logistic growth for the prey including hunting cooperation proposed in [1] has been extended by assuming that both prey and predators can linearly spread in the environment. Linear stability analysis has been performed and conditions guaranteeing that a biological meaningful equilibrium, stable in the absence of diffusion, becomes unstable in the presence of diffusion (Turing instability) have been found. For this model, we showed some numerical simulations which highlight the impact of different choices for the diffusion coefficients on the dynamics of the two populations and in particular we explored the Turing patterns formation. In the present paper, which can be seen as an enrichment of [7], we analyze again the problem with the aim of showing destabilization mechanisms of the biologically meaning equilibrium and detecting aspects for the eventual oscillatory pattern formation. Moreover, in order to complete the stability analysis of the coexistence equilibrium, we show a nonlinear stability result. The plan of the paper is as follows. Section 2 introduces the mathematical model while a nonlinear stability result is performed in Section 3.InSection 4 linear stability/instability results, both when populations are homogeneously mixed in the environment and in the spatial dependent case, are presented and extensive numerical simulations, depicting the theoretical aspects reported in the manuscript, are showed.

2 Mathematical model

In this paper we reconsider the predator–prey model proposed in [7] concerning hunting cooperation with Allee effect on the predators. Assuming that both prey and predators can (linearly) diffuse in the environment, the model, in nondimensional form, that we investigate is given by 𝜕 ( ) ⎧ n 𝜎 n 𝛼 𝛾 , ⎪ 𝜕 = n 1 − − (1 + p)np + 1Δn ⎨ t k (1) 𝜕p ⎪ = (1 + 𝛼p)np − p + 𝛾 Δp, ⎩ 𝜕t 2 where the nondimensional variables are e𝜆 𝜆 x n = N, p = P, t = m𝜏, X = , (2) m m A and r e𝜆K̄ am d 𝜎 = , k = ,𝛼= ,𝛾= i , i = 1, 2. (3) m m 𝜆2 i A2m In particular, N(X,𝜏)andP(X,𝜏) denote, respectively, the prey and predator densities; for any 𝜑 ∈ {N, P},𝜑:(X,𝜏) ∈Ω×ℝ+ → 𝜑(X,𝜏) ∈ ℝ+ being Ω a bounded, connected, and open subset of ℝn (n = 2, 3) having the internal cone propriety; A is the diameter of Ω. The (positive) constants appearing in (1)–(3) have the following biological meaning: r is the per capita intrinsic growth rate of prey; K̄ is the carrying capacity of prey; e is the food conversion efficiency; m is the per capita mortality rate of predators; 𝜆 is the attack rate; F. Capone et al.: On a reaction–diffusion system modelling Allee effect on predators | 3

a is the predator cooperation in hunting rate; and di are the diffusion coefficients. We append the following initial-boundary conditions { ∇n ⋅ n = 0, ∇p ⋅ n = 0, on 𝜕Ω×ℝ+. , , , , . (4) n(x 0) = n0(x) p(x 0) = p0(x) on Ω being n the outward unit normal to 𝜕Ω. , The biologically meaningful constant equilibria of system (1) are: (1) E0 = (0 0) (null solution); (2) , ∗ ∗, ∗ E1 = (k 0) (prey-only equilibrium); (3) E = (n p ) (coexistence equilibrium), with 1 n∗ = (5) 1 + 𝛼p∗ and p∗ given by the positive solution of the equation [7]

k𝛼2x3 + 2𝛼kx2 + k(1 − 𝜎𝛼)x + 𝜎(1 − k) = 0. (6)

In order to perform the stability analysis for the coexistence equilibrium E∗ we set

X = n − n∗, Y = p − p∗. (7)

System (1) becomes ( ) ( )( ) ( ) ( ) 𝜕 𝛾 X a11 a12 X 1ΔX F1 , 𝜕 = + 𝛾 + (8) t Y a21 a22 Y 2ΔY F2 with ⎧ 𝜎 1 + 2𝛼p∗ a =− , a =− , ⎪ 11 𝛼 ∗ 12 𝛼 ∗ ⎪ k(1 + p ) 1 + p 𝛼p∗ ⎪a = (1 + 𝛼p∗)p∗, a = , ⎨ 21 22 1 + 𝛼p∗ (9) ⎪ 𝜎X2 F =− − (1 + 𝛼p∗)XY − 𝛼Y(XY + n∗Y + p∗X), ⎪ 1 k ⎪ 𝛼 ∗ 𝛼 ∗ ∗ . ⎩F2 = (1 + p )XY + Y(XY + n Y + p X) To (8) we associate the boundary conditions

∇X ⋅ n =∇Y ⋅ n = 0, on 𝜕Ω×ℝ+ (10) andwedenotebyW∗(Ω) the functional space defined by { } , , d𝜑 W∗(Ω)= 𝜑 ∈ W1 2(Ω) ∩ W1 2(𝜕Ω): =0on𝜕Ω×R+ . dn

Let 𝛼 <𝛼 <𝛼 < <𝛼 < 0 = 0 1 2 ··· n ··· (11) be the eigenvalues of the operator −Δ on Ω with homogeneous Neumann boundary conditions. The linear system linked to (8) is given by 𝜕X = X, (12) 𝜕t where ( ) ( ) 𝛾  a11 + 1Δ a12 , X . = 𝛾 X = (13) a21 a22 + 2Δ Y For each i = 0, 1, 2, 3, …, 𝜉 is an eigenvalue of  if and only if 𝜉 is an eigenvalue of the matrix (see [18]) ( ) a − 𝛼 𝛾 a ̃ 11 i 1 12 . (14) i = 𝛼 𝛾 a21 a22 − i 2 4 | F. Capone et al.: On a reaction–diffusion system modelling Allee effect on predators

̃ The characteristic equation of i is 𝜆2 𝜆 , i − I1i i + I2i = 0 (15) , , ̃ with Iji (j = 1 2), the principal i− invariants: ⎧ 0 𝛼 𝛾 𝛾 , ⎪I1i = I1 − i( 1 + 2) 𝜎𝛾 𝛼 ∗𝛾 (16) ⎨ 𝛾 𝛾 𝛼2 2 − k p 1 𝛼 0, ⎪I2i = 1 2 i + i + I2 ⎩ k(1 + 𝛼p∗)

0, , where I j (j = 1 2), are the principal invariants in absence of diffusion, i.e. k𝛼p∗ − 𝜎 ⎧I0 = a + a = , ⎪ 1 11 22 𝛼 ∗ k(1 + p[) ] ⎨ 𝜎𝛼 (17) ⎪I0 = a a − a a = p∗ − + 1 + 2𝛼p∗ . ⎩ 2 11 22 12 21 k(1 + 𝛼p∗)2

The necessary and sufficient conditions guaranteeing that all the roots of (15) have negative real part are the Routh–Hurwitz conditions [19] < , > , , , , . I1i 0 I2i 0 ∀i = 0 1 2 … (18)

3 Nonlinear stability

Setting 𝜇 , 𝜇 , X = 1U1 Y = 2U2 (19) 𝜇 , , being i (i = 1 2) two positive scalings to be suitably chosen, (8) can be written as 𝜕 ⎧ U1 𝛾 1 ̃ , ⎪ 𝜕 = b11U1 + b12U2 + 1ΔU1 + 𝜇 F1 t 1 ⎨𝜕 (20) ⎪ U2 𝛾 1 ̃ , ⎩ 𝜕 = b21U1 + b22U2 + 2ΔU2 + 𝜇 F2 t 2 with 𝜇 j , ̃ (𝜇 ,𝜇 ). (21) bij = aij 𝜇 Fi = Fi 1U1 2U2 i In this section we perform the nonlinear stability analysis of the coexistence equilibrium with respect to the norm [20] [ 1 0 ‖ ‖2 ‖ ‖2 ‖ ‖2 V = I (‖U ‖ + ‖U ‖ ) + ‖b U − b U ‖ 2 2 1 2 11 2 21 1 ] ‖ ‖2 . + ‖b12U2 − b22U1‖ (22)

ThetimederivativeofV, evaluated along the solutions of (20),isgivenby

̇ 0 0 ‖ ‖2 ‖ ‖2 , V = I1 I2 (‖U1‖ + ‖U2‖ ) +Φ1 +Φ2 (23) being ⎧ 0 2 2 , 0 2 2 , , A1 = I2 + b21 + b22 A2 = I2 + b11 + b12 A3 = b11b21 + b12b22 ⎪ ⟨ ,𝛾 ⟩ ⟨ ,𝛾 ⟩, ⎨Φ1 = A1U1 − A3U2 1ΔU1 + A2U2 − A3U1 2ΔU2 (24) ⎪ 1 ⟨ , ̃ ⟩ 1 ⟨ , ̃ ⟩. ⎩Φ2 = 𝜇 A1U1 − A3U2 F1 + 𝜇 A2U2 − A3U1 F2 1 2

Lemma 1. If √ 𝛾 𝛾 | | ≤ 𝜀 | |𝛾 𝛾 , <𝜀< , ( 1 + 2) A3 2(1 − ) A1A2 1 2 0 1 (25) F. Capone et al.: On a reaction–diffusion system modelling Allee effect on predators | 5 then ≤ 𝜀 𝛾 ‖ ‖2 𝜀 𝛾 ‖ ‖2 . Φ1 − A1 1 ‖∇U1‖ − A2 2 ‖∇U2‖ (26)

Proof. From (24)4, it follows that 𝜀 𝛾 ‖ ‖2 𝜀 𝛾 ‖ ‖2 𝛾 𝛾 ⟨ , ⟩ Φ1 =−(1 − )A1 1 ‖∇U1‖ − (1 − )A2 2 ‖∇U2‖ + A3( 1 + 2) ∇U1 ∇U2 𝜀 𝛾 ‖ ‖2 𝜀 𝛾 ‖ ‖2 . − A1 1 ‖∇U1‖ − A2 2 ‖∇U2‖ (27)

Then, if (25) holds true, the thesis is reached.

Lemma 2. Denoting by K(Ω) a positive constant depending on the domain Ω, it follows that

2 2 1 ≤ ‖ ‖ ‖ ‖ 2 ⋅ Φ2 K(Ω)(‖U1‖ + ‖U2‖ ) ⋅ ‖ ‖2 ‖ ‖2 ‖ ‖2 ‖ ‖2 . (‖U1‖ + ‖U2‖ + ‖∇U1‖ + ‖∇U2‖ ) (28)

Proof. As we have proved in [7], there exist two positive constants, M1 and M2 such that ‖ ‖ ≤ , ‖ ‖ ≤ . X ∞ M1 Y ∞ M2 (29)

Setting | | | | |A 𝜎𝜇 | | A 2 | c = | 1 1 | , c = |A 𝛼n∗𝜇 + 3 𝛼𝜇 n∗| , 1 | k | 2 | 2 2 𝜇 2 | | 1 | | A 𝜇 A 𝜎𝜇 | c = |−(1 + 𝛼p∗)A − 𝛼𝜇 p∗A − 3 1 (1 + 𝛼p∗) − A 𝛼p∗𝜇 + 3 1 | , 3 | 1 2 1 𝜇 3 1 k | | 2 | ((30)) | A 𝛼n∗𝜇2 | | 1 2 𝛼 ∗𝜇 𝛼 ∗ 𝜇 𝛼 ∗𝜇 𝛼 ∗ 𝜇 ∗𝛼𝜇 | , c4 =|− 𝜇 + A2 p 1 + A2(1 + p ) 1 − A3 n 2 + A3(1 + p ) 2 + A3 p 2| | 1 | | | | | | 𝛼𝜇2 𝛼𝜇 𝜇 | , | 𝛼𝜇 𝜇 𝛼𝜇2| , c5 = |A1 2 + A3 1 2| c6 = |A2 1 2 + A3 2| one has that

≤ ⟨| |, 2⟩ ⟨| |, 2⟩ ⟨| |, 2⟩ Φ2 c1 U1 U1 + c2 U2 U2 + c3 U2 U1 ⟨| |, 2⟩ ⟨ 2, 2⟩ ⟨| |, | |3⟩. + c4 U1 U2 + c5 U1 U2 + c6 U1 U2 (31) In view of the Holder inequality and the boundedness of solutions, it turns out that ( ) 2M c Φ ≤ c ‖U ‖ ‖U ‖2 + c + 1 6 ‖U ‖ ‖U ‖2 2 1 ‖ 1‖ ‖ 1‖4 2 𝜇 ‖ 2‖ ‖ 2‖4 1 ( ) 2M c + c + 1 5 ‖U ‖ ‖U ‖2 + c ‖U ‖ ‖U ‖2 . (32) 4 𝜇 ‖ 1‖ ‖ 2‖4 3 ‖ 2‖ ‖ 1‖4 1 Applying the Sobolev inequality

‖ ‖2 ≤ ‖ ‖2 ‖ ‖2 , f 4 c(Ω)( f + ∇ f ) one obtains ≤ ̃ ‖ ‖ ‖ ‖ ‖ ‖2 ‖ ‖2 ‖ ‖2 ‖ ‖2 Φ2 k(Ω)(‖U1‖ + ‖U2‖)(‖U1‖ + ‖U2‖ + ‖∇U1‖ + ‖∇U2‖ ) (33) and the thesis is reached in view of the inequality √ 2 2 1 ‖ ‖ ‖ ‖ ≤ ‖ ‖ ‖ ‖ 2 . ‖U1‖ + ‖U2‖ 2(‖U1‖ + ‖U2‖ ) (34)

Theorem 1. If (25) holds together with the conditions guaranteeing the linear stability of the coexistence equilibrium, then this equilibrium is nonlinearly, asymptotically, locally, exponentially stable in the V-norm. 6 | F. Capone et al.: On a reaction–diffusion system modelling Allee effect on predators

Proof. In view of (26) and (28), it follows that

̇ ≤ | 0 0| ‖ ‖2 ‖ ‖2 𝜀 𝛾 ‖ ‖2 𝜀 𝛾 ‖ ‖2 V − I1 I2 (‖U1‖ + ‖U2‖ ) − A1 1 ‖∇U1‖ − A2 2 ‖∇U2‖ 2 2 1 2 2 2 2 ‖ ‖ ‖ ‖ 2 ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ . + K(Ω)(‖U1‖ + ‖U2‖ ) (‖U1‖ + ‖U2‖ + ‖∇U1‖ + ‖∇U2‖ ) (35)

Setting { } { } 1 |I0| |I0| q = min , 2 , q = max 2 + b2 + b2 + b2 + b2 , (36) 1 2 2 2 2 11 12 21 22 one has that ‖ ‖2 ‖ ‖2 ≤ ≤ ‖ ‖2 ‖ ‖2 . q1(‖U1‖ + ‖U2‖ ) V q2(‖U1‖ + ‖U2‖ ) (37) Then ( ) 0 0 | | 1 ̇ I1 I2 K(Ω) V ≤ − − √ V 2 V+ q2 q1 q1 ( ) K(Ω) 1 2 2 𝛾 √ 2 ‖ ‖ ‖ ‖ , − − V (‖∇U1‖ + ‖∇U2‖ ) (38) q1 with 𝛾 𝜀 𝛾 ,𝜀 𝛾 . = min { A1 1 A2 2} (39)

If { √ √ } 1 q q |I0I0| 𝛾 q 2 ≤ 1 1 1 2 , 1 , V0 min (40) q2K(Ω) K(Ω) by recursive arguments, V̇ < 0, i.e. the coexistence equilibrium is noninearly (locally), asymptotically, exponentially, stable in the V-norm.

4 Linear stability/instability results: numerical simulations

The aim of this section is to show linear stability/instability results, both when populations are homogeneously mixed in the environment and in the spatial dependent case in order to highlight destabilization mechanisms of the biologically meaning equilibrium and detect aspects for the eventual oscillatory pattern formation. Extensive numerical simulations, depicting the theoretical aspects, are showed. In all the reported examples we fixed the per capita growth rate of preyas 𝜎 = 3. Linear stability analysis in the spatially uniform case highlights that E∗ loses its stability by two types of bifurcations. One is the Hopf bifurcation, the other is the transcritical one. The Hopf bifurcation is characterized by the fact that the equilibrium E∗ loses its stability since a couple of complex conjugate eigenvalues 0 ∗ 0 ∗ > crosses the imaginary axis. The Hopf bifurcation can be detected by requiring I1 (E ) = 0andI2 (E ) 0, that is, for the model at hand, 𝜎 = k𝛼p∗ (41) and k(1 + 𝛼p∗)2(1 + 2𝛼p∗) >𝛼𝜎. (42) ∗ 0 ∗ < 0 ∗ > Precisely, the spatially homogeneous equilibrium E ,(linearly)stableforI1 (E ) 0andI2 (E ) 0 I0 ∗ loses its stability when 1 (E ) = 0 and a limit cycle can arise surrounding√ the unstable steady state for < 0 ∗ ≪ 𝜋 𝜔 𝜔 0 0 I1 (E ) 1. The periodic solution has period T = 2 ∕ ,where = I2 is the angular frequency. Assuming weak hunting cooperation among predators (𝛼 = 0.3) and high basic reproduction number for a homogeneously mixed predator population (k = 5), a single and stable coexistence equilibrium exists, given F. Capone et al.: On a reaction–diffusion system modelling Allee effect on predators | 7

∗ , by E ≈ (0.66, 1.72). Figure 1 shows results of the simulation started at (n0 p0) = (0.7, 1.8): the phase plane plot, where the stable coexistence equilibrium can be clearly seen, and trajectories of both populations. When we choose a very low basic reproduction number (k = 0.8) and a strong hunting cooperation (𝛼 = 10) the coexistence equilibrium E∗ is oscillatory. Limit cycles emerge, as shown in Figure 2 where, , starting from the initial conditions (n0 p0) = (0.2, 0.5), the limit cycle in the phase plane is plotted (left panel). The right panel of the same figure shows the time evolution of both populations. The other bifurcation is the transcritical one, characterized by the fact that the equilibrium E∗ loses its 0 ∗ stability since one real eigenvalue vanishes. We find this kind of bifurcation by solving I2 (E ) = 0, i.e. k(1 + 𝛼p∗)2(1 + 2𝛼p∗) − 𝛼𝜎 = 0. (43)

When populations are heterogeneously mixed in the environment, the linear stability analysis shows the possibility of diffusion-driven instability (Turing instability), that is responsible for the eventual emergence of spatial patterns for system. We recall that a reaction–diffusion system exhibits Turing instability if a steady state, which is stable in the absence of diffusion, becomes unstable to small spatial perturbations inthe presence of diffusion. In7 [ ] we found both necessary conditions and sufficient conditions for the occurrence of Turing instability. These results define a region in the parameter space in which the formation of stationary spatial patterns is hence expected. The points in which different bifurcation curves meet correspond to bifurcations of higher codimension. The knowledge about the occurrence of higher codimension bifurcations could be used to draw conclusions about the possibility of complex spatio-temporal dynamics ([21, 22]). Precisely, following the analysis showed in [17], we highlight that the transcritical and the Turing bifurcation curves meet at the point transcritical–Turing (TT), whose coordinates satisfy (43) and Turing instability conditions. Moreover, the two other interesting points Takens–Bogdanov (TB) and Turing–Hopf (TH) in the bifurcation diagram (for the potential onset of complex dynamics) are – Point TB, given by the intersection of the transcritical and Hopf lines (in this case both eigenvalues are equal to zero). This indicates the occurrence of a Takens–Bogdanov bifurcation; – Point TH, given by the intersection of the Hopf and Turing bifurcation curves. In the neighbor of this point, the formation of inhomogeneous stationary patterns caused by Turing instabilities competes with

Figure 1: Numerical simulation of the reaction part of system (1) with parameters choice 𝛼 = 0.3; k = 5; 𝜎 = 3. In the left panel, the phase plane plot with the stable coexistence equilibrium; in the right panel, trajectory of the prey (dark gray) and the predators (light gray).

Figure 2: Numerical simulation of system (1) in absence of diffusion with parameter choice 𝛼 = 10; k = 0.8; 𝜎 = 3. In the left panel, plot of the limit cycle in the phase plane; in the right panel, trajectory of the prey (dark gray) and the predators (light gray). 8 | F. Capone et al.: On a reaction–diffusion system modelling Allee effect on predators

the onset of homogeneous oscillations due to a Hopf bifurcation, leading to the possible emergence of an interesting class of spatio-temporal behaviors.

< 0 ≪ Following [17], when the reaction part of system (1) admits a limit cycle via a Hopf bifurcation (0 I1 1), < , < if I1i 0 I2i 0 then Turing–Hopf instabilities appear and they are called weak TH instabilities. Weak instabilities are depicted as slight oscillations with the frequency of the cycle solution superposed over a prevalent inhomogeneous pattern. These oscillations become greater with the increment of the amplitude of 0 the limit cycle i.e. with the increment of I1 . The region around a Turing–Hopf bifurcation is of most interest for our study and we perform simulations for the parameter sets of interest. As widely reported in the literature [23], to see the instability patterns in the solution one has to choose a sufficiently large size for the space domain with respect to the wavelength of the instable modes. Tocomply with this request, we set as space domain the interval [0, 10] for 1D simulations, or the square [0, 10] × [0, 10] for 2D simulations, with a discretization step h = 0.05. Moreover, we assume an initial condition very close to the equilibrium value plus a small random perturbation. Specifically, the initial values for the two , populations are fixed as (n0 p0) = (0.7, 1.8) plus a random spatial perturbation of amplitude 0.01 on every point of the numerical grid. If the diffusion coefficients are chosen in the stability range, by setting for example 𝛾 𝛾 1 = 2 = 0.1, the considered equilibrium point remains stable, both in the 1D setting (not shown) and in the considered 2D setting: the state of the prey and predator populations are represented in Figure 3:both populations rapidly approach the equilibrium state and the fluctuations in the solution are quickly damped out On the contrary, if the diffusion coefficients are chosen in the range guaranteeing Turing instability, the system evolves with a different dynamics, since the coexistence equilibrium loses stability due to the Turing effect and spatial patterns emerge in both populations (see Figure 4). The insurgence of Turing patterns has been already considered in [7], but we choose to include it in this work to give a comprehensive description of the dynamics of system (1) under different parameters settings.

𝛾 𝛾 Figure 3: Surface plot of the state of the system for T = 500 in presence of (balanced) diffusion: 1 = 2 = 0.1. Note that both populations (prey, in the left panel; predators in the right panel) have reached the equilibrium value: The random fluctuations on the initial state are damped and oscillations in the solution are below the reported decimal digits.

𝛾 𝛾 Figure 4: Surface plot of the state of the system for T = 500 in presence of (unbalanced) diffusion: 1 = 1.5; 2 = 0.01. Note that both populations (prey, in the left panel; predators in the right panel) show spatial patterns of great amplitude. F. Capone et al.: On a reaction–diffusion system modelling Allee effect on predators | 9

𝛾 𝛾 𝛼 Figure 5: Numerical simulation of system (1) in presence of ‘‘balanced’’ diffusion, 1 = 2 = 0.1, with parameter choice = 10; k = 0.8; and 𝜎 = 3. Complete trajectory of the prey population (left) and enlarged view of the final part (right).

𝛾 𝛾 𝛼 Figure 6: Numerical simulation of system (1) in presence of ‘‘balanced’’ diffusion, 1 = 2 = 0.1, with parameter choice = 10; k = 0.8; and 𝜎 = 3. Complete trajectory of the predator population (left) and enlarged view of the final part (right).

𝛾 𝛾 𝛼 Figure 7: Numerical simulation of system (1) in presence of ‘‘unbalanced’’ diffusion, 1 = 1.5, 2 = 0.01, with parameter choice = 10; k = 0.8; and 𝜎 = 3 as in the previous figure. Complete trajectories of the prey (left) and predator (right) populations for T = 500.

Let us choose again the parameters value as in Figure 2 and simulate the corresponding model with diffusion (1). Again, experiments have been performed in both the 1D and 2D settings, but we prefer to report just the results referring to the 1D setting to see the complete trajectories of the involved populations. Indeed, analogous plots for the 2D setting could have been produced only at some single fixed time instants. 𝛾 𝛾 First, we consider “balanced” diffusion ( 1 = 2 = 0.1). In this case, the same instability observed in the ODE model persists and the system still shows an oscillatory behavior: Figures 5 and 6 report the trajectories for the prey and predator populations, respectively, up to T = 500. 𝛾 𝛾 On the other hand, if we choose “unbalanced” diffusion ( 1 = 1.5, 2 = 0.01) the model shows a very different instability pattern, as reported in Figure 7. Several studies have explored this simultaneous appearance of Hopf bifurcation and Turing instabilities (Turing–Hopf bifurcations) in different scenarios (see for instance [24] and reference therein). In our opinion, it is worth deserving a particular attention to these numerical results in view of the numerous applications involving biological processes, ecosystem evolution, perma- nence, and control of populations. In fact, as the numerical simulations highlight, diffusion can strongly influence the fate of ecosystems.

Acknowledgement: The paper has been performed under the auspices of GNFM and GNCS of INdAM and partially supported by the Regione Campania Project REMIAM. 10 | F. Capone et al.: On a reaction–diffusion system modelling Allee effect on predators

Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission. Research funding: None declared. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

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