A SIMPLE MODEL FOR BIOLOGICAL AGGREGATION WITH ASYMMETRIC SENSING

Transcription

1 A SIMPLE MODEL FOR BIOLOGICAL AGGREGATION WITH ASYMMETRIC SENSING PAUL A. MILEWSKI, XU YANG Abstract. In this paper we introduce a simple continuum model or swarming o organisms in which there is a nonlocal aggregation term with an asymmetric sensing kernel countered by a nonlinear diusion. The model can be thought o as capturing the dynamics o organisms that aggregate responding to visual cues and whose ield o vision is predominantly in the direction o motion. The model has a variety o traveling solutions, including compact swarms where the speed o the swarm increases with the total number o organisms in the swarm (up to a largest swarm), traveling ronts and periodic waves. The dynamics o ully time-dependent solutions include complicated inelastic swarm interactions and the spontaneous breakup o small populations into compact swarms. We also introduce a bi-directional model where the organisms are allowed to change directions. Key words. swarming, chemotaxis, travelling waves. AMS subject classiications.. Introduction. There exist a wide variety o models or the aggregation o organisms. These range rom modeling the chemotaxis o bacteria, where the organisms sense variations in an ambient chemical attractant or repellent, to the social aggregation o higher organisms [, 8]. Here, we ocus on an idealized simple continuum model or aggregation, and the travelling compact, periodic and ront solutions resulting rom the nonlocal sensing o nearby populations. We do not consider precise biological details on how individual behavior results in continuum aggregation-diusion models (see [,, 7, 6]), since a goal is to analyze complex behavior arising rom the simple equations. The model involves directional sensing, that is, or example when the organism s response is to visual cues and the ield o vision is predominantly in a particular direction, such as the organism s direction o motion. Such a nonlocal response rom visual cues can be an important mechanism or the behavior o ungulate herds, schools o ish and insect swarms [4, 5, 9]. Directional sensing may also be applicable in modelling the more detailed dynamics o certain chemotactic processes where a subset o the population is moving and sensing only in a particular direction. For example, swimming bacteria have a memory o the spatial concentration o a chemoattractant behind them. We irst introduce the simple one-dimensional one-way model, in which the sensing is always in a ixed direction and organisms move preerentially in the direction they are sensing. The model allows or analytical analysis o the resulting solutions. We ind that the model supports a one parameter amily o compact traveling waves or swarms whose speed increases with the swarm size with a limiting, maximum sized, swarm. (Periodic arrays o swarms are also ound.) In addition, the model has traveling ront solutions, where a uniorm large population level migrates into a region o lower population (or, singular swarms travelling into a region o zero population) and periodic and solitary waves o organisms about a constant nonzero level. Using analysis and asymptotic arguments we characterize these traveling solutions and discuss their stability. We then perorm numerical simulations o this simple one-way model in a periodic Department o Mathematics, University o Wisconsin, 48 Lincoln Dr., Madison, WI 5376 (milewski@math.wisc.edu). Paul Milewski is partially supported by NSF-DMS

2 2 PAUL MILEWSKI, XU YANG domain, and ind that suiciently small initial data will spontaneously aggregate into swarms. Larger data will result in periodic solutions. We also ind that swarms o dierent size can coalesce in collisions that result in larger swarms together with some radiation (organisms let behind). We conclude by introducing a bi-directional model in which a particular organism may change the direction in which it is sensing according to transition probabilities that depend on, or example, local population densities. This type o model is more realistic and could have applications in modeling the run/tumble transitions in certain chemotactic processes. 2. Proposed Model. Consider a scalar conservation law model or the density o organisms ρ(x, t) o the orm (2.) ρ t + (ρ u(ρ)) x = where the speed o the organisms u is modeled as having two contributions: a nonlocal attraction which will tend to nucleate swarms, together with a local repulsion that prevents the densities o organisms to grow too large. We shall choose in this paper, similarly to some models discussed in [5], (2.2) u = K ρ ρρ x with the kernel K deined by (2.3) K = { e x, x, x > The convolution is deined as K ρ = K(ξ)ρ(x ξ) dξ and, since K has support or ξ, the result o the convolution term is that a contribution to the speed o the organisms at a particular point is proportional to a weighted average o the organisms to the right o that point. Thus we are modeling right-sensing organisms, and an increasing population density to the right o an organism will tend to increase its locomotion speed. The nonlinear term in (2.2), proportional to the gradient o ρ, gives a contribution to the speed that is in the opposite direction o increasing ρ and thereore leads to repulsion o nearby organisms. Even though we chose a particular kernel or its simplicity in the analysis, it is expected (based on a ew numerical experiments) that a wide class o asymmetric Kernels could be used with similar results. We emphasize that the aggregation term is non-local and senses organisms at a distance O() away rom a particular point whereas the repulsion term is local. We have chosen all parameters to be one in (2.2) without loss o generality: or u = αk(βx) ρ γρρ x the parameters could be eliminated by a rescaling o x, ρ and t. Furthermore, higher powers o ρ in the nonlinear diusive term are also amenable to the type o analysis we show here. In order to gain intuition on the dynamics and compare with a similar model with a symetric kernel [2], it is useul to consider the aggregation term in Fourier space. Then, F {K ρ} = ( ik ρ = + k 2 + ik ) + k 2 ρ = F {( + x )S ρ}, where F(ρ) = ρ is the Fourier transorm and k the dual variable to x. S = 2 exp( x ) is the Poisson kernel whose Fourier transorm is Ŝ = +k. This kernel is the symmetric aggregation kernel used in 2 [2].

3 Simple Model or Aggregation 3 In terms o S, our model could be written as (2.4) ρ t + [ρ(s ρ)] x + [ ρ(s ρ) x ρ 2 ρ x ]x =, and, i the second term above is removed, one recovers the model in [2]. In their model, the dynamics is much simpler with all initial data tending to localized steady solutions. The additional term in our model clearly breaks the symetry and allows or traveling solutions. It is useul to interpret physically the two nonlocal terms in (2.4). The term [ρ(s ρ) x ] x is anti-diusive and gives a contribution to the speed o the organisms as proportional to the gradient o a locally symmetrically averaged population, resulting in aggregation. The term [ρ(s ρ)] x is advective and gives a positive contribution to the speed proportional to the symmetrically averaged population. This aspect o the model is probably applicable only to a restricted class o organisms, however, it has been observed that in ish schools, the speed o the ish increase with the density o the school [3]. A urther interpretation o the model is seen by writing the equation as a system or ρ(x, t) and the local average population w(x, t) = S ρ ρ t + [wρ] x = [ ρ 2 ρ x ρw x ]x, w xx = (ρ w). Here it is clear that w is a smoothed density which controls the advection and aggregation terms in the equation or ρ. At this point, we can make an analogy with chemotaxis. In certain swarming phenomena usually described by Keller-Segel type equations [2], bacteria swim up gradients in chemoattractants released by the bacteria themselves. However, at the bacterial level, the sensing o the chemoattractant and depends on a run/tumble dynamics: indivitual bacteria swim and then tumble to change directions randomly and their run/tumble requency depends on the sensed changes in concentration o attractant. In the equations above, one could associate ρ with the right moving bacteria and w to the chemoattractant. In order to complete a reasonable model one would have to include let moving bacteria with transitions between the two directions corresponding to a tumble (see section 5). This bidirectional equation would be akin to a telegraph equation approximation o chemotactic random walks. 3. Traveling waves. In this section we study the traveling wave solutions to our model, both rom a direct analysis o the resulting equation under the traveling wave assumption and using phase plane methods. There are two types o solutions: singular solutions where the population density is zero somewhere in the domain and nonsingular solutions which are nowhere zero. There are many types o traveling waves: swarms with compact support, rareaction ronts (corresponding to a decrease in population density with the passage o the ront) and invasion ronts (corresponding to an increase in population density), periodic waves and noncompact solitary waves on a nonzero background. Invasion ronts are urther divided into 2 types: those traveling into regions where ρ = ahead o the ront and those traveling into regions o lower but nonzero uniorm population. Periodic solutions also come in two types: those consisting o smooth solutions where everywhere ρ > and those consisting o a periodic array o compact swarms. Initially we restrict our discussion to solutions which propagate on the zero state such as a single swarm or a ront propagating into the zero state. We then discuss

4 4 PAUL MILEWSKI, XU YANG ronts between two constant (nonzero) states, periodic, and noncompact solitary solutions. Periodic solutions are in act the ones we simulate in our time dependent computations since these are perormed on a periodic domain. In order to study the traveling wave solutions we deine (3.) or equivalently (3.2) v K ρ = e ξ ρ(x + ξ)dξ, v x = v ρ, v() = v, and seek solutions u = (θ) with θ = x ct. Then, (2.) becomes (3.3) c + (v) 3 (3 ) =. 3.. Singular solutions: compact swarms and singular ronts. Consider solutions where the population is zero or θ > L We will show that these solutions are either compactly supported swarms or ronts traveling into a region where ρ =. Then, integrating once and assuming that the constant o integration is zero (which can be veriied as consistent a posteriori), dividing by, and integrating the result on an interval θ [, L], we obtain (3.4) cl + v dθ 2 ( 2 (L) 2 () ) =. Thus, i the solution is compactly supported on [, L] with () = (L) = and we deine the total mass o the swarm M by: (3.5) we then have (3.6) c = L M = dθ, v dθ = L (M v ). We have used the act that v = v, v(l) = and v = v(). In addition, rom (3.3), carrying out similar steps without dividing by we can obtain: c = ( L ) v dθ = L (3.7) v 2 dθ + M M 2 v2, We irst derive bounds on the speed c. Using the Hölder inequality v2 dθ, in (3.7), and eliminating v using (3.6) we get L (3.8) M L + 2 c M L ( ) L 2 L vdθ We will sharpen the upper bound (see below) when looking at the solutions in the phase plane (, v).

5 Simple Model or Aggregation (c,c).3 v,v θ Fig. 3.. Phase plane diagram (let) and traveling swarm solutions (right). In the phase plane, the dashed lines are the nullclines. Both and v are shown. For this example c =.2. (3.9) (3.) The phase plane, v, and or a ixed c has solutions satisying = v c, v = v. Typical solutions are shown in Figures 3. and 3.2. The low has an unstable ixed point (spiral) at (c, c) and is singular when =. This singularity allows or compact solutions o the orm shown in the next section. Not every solution curve in the phase plane corresponds to physical traveling solutions o (2.). In addition to the ixed point, which corresponds to the constant solution (see Section 4 or the stability o this solution), only solutions limiting to (, ) as θ L are relevant. This is because, i a solution has = or θ L, then v(l) must be zero by (3.). As we shall see, this leaves two kinds o possible solutions: corresponding to swarms (compact solitary waves) and to ronts into the zero state (singular ronts). Swarm solutions are deined by the phase plane trajectory or θ (, L) that begin and terminate on the line = connecting the points (, v ) and (, ). For these solutions = or θ L and θ, v = or θ L, but v or θ (v decays exponentially as θ, by (3.)). Figure 3. shows an example o such a solution. Front solutions are deined by trajectories in the phase plane or θ (, ) that terminate at (, ) or θ = but tend asymptotically to (c, c) as θ. For θ >, = v =. Figure 3.2 shows an example o such a solution. O course, either the ront or swarm solutions can be shited arbitrarily in x. For a given c there is either a swarm or a ront solution. Swarm solutions exist or < c c whereas ront solutions exist or c > c. Clearly, rom the phase plane or swarm solutions to exist, v c so that >. Physically, v is the attractive part o the speed o the last organism due to the organisms ahead o it. Given that organisms will also eel some repulsion due to local density gradients, v needs to be larger than c i the last organism is to remain in the swarm and not all behind. We shall see below that or vanishingly small swarms, v = 2c, and that v /c decreases as c increases. Then, we deine c, the astest swarm speed, as the value o c with a swarm solution or which v /c =. That is, the speed c is such so that the swarm

6 6 PAUL MILEWSKI, XU YANG v v.2,v θ Fig Phase plane diagram (let) and traveling singular ront solutions (right). In the phase plane, the dashed lines are the nullclines. Both and v are shown. For this example c =.25. trajectory in the phase plane is between (, c) at θ = and (, ) at θ = L. We can now sharpen the upper bound or the swarm speed since v c implies, using (3.6), that c M/(L + ). Thus we have that (3.) M L + 2 c M L + To understand the transition rom swarm solutions to ront solutions at c = c, it is easier to think o by reversing time in the phase plane, turning (c, c) into a stable spiral. Then we can ask what happens to solutions starting at (, ) (with the right asymptotics as derived in Section 3.2). For c < c they terminate at = above v = c corresponding to the compact swarms. For c > c they cross the nullcline v = c or > and spiral inwards to the ixed point (c, c). These are traveling invasive ronts, corresponding to organisms traveling into an empty region. The ronts are not monotonic, having a decaying oscillatory component (see Figure 3.2 ). We ind numerically that c.233. In Figure 3.4 we show the comparison between the bounds (3.), the computed values o M/c, and an approximation derived in Section 3.2 or < c < c. Note that the bounds appear sharp. From time dependent numerical computations it seems that individual solitary swarms are stable. Collisions between swarms may have complex behavior (see Section 4). Constant solutions, however, are only stable at large amplitude (see Section 4) and thereore one expects the ront solutions to be unstable unless the constant state behind the ront is suiciently large. We conclude this section with two observations. First, in time dependent computations, we shall consider periodic solutions (with period l). Thus, we shall be computing periodic arrays o travelling swarms. These can be computed in the phase plane also: denoting ρ l the unction that is equal to the period l unction ρ(x ct) on [, l] and otherwise (and assuming that l > L where L is the length o the swarm), then the nonlocal term is modiied v = K ρ = K ρ l + e (l θ) e l vl K ρ l + q(x; l, v l ).

7 Simple Model or Aggregation ,v.25,v θ θ Fig Comparison o isolated and periodic (curves with ) swarms or the same speed c =.2. Both and v are shown. Let: l = 5, right: l = 8. Thus the equations or the single copy o the periodic swarm are (3.2) (3.3) l = vl + q(x; l, v l ) c l, v l = v l l. Given that the system now has v l as a parameter, it must now be solved by an iterative scheme (we used a simple shooting method). Figure 3.3 compares the ininite domain with the periodic domain solutions o the same speed c in the ininite and in the periodic case. We see that the periodic solution has slightly smaller mass which is intuitively reasonable given that it is sensing the swarms ahead o it. Also, the igure shows that as l increases, the periodic swarms tends rapidly to the isolated one. A second observation is that, numerically, the phase plane pictures are computed by transorming the equations (3.) using g = 2 /2 and reversing time to give (3.4) (3.5) g = v + c, v = v + 2g. The initial data is (g, v)() = (, ) and the trajectories either reach (, v ) or spiral to (c 2 /2, c). This transormation desingularizes = and is more amenable to numerics Asymptotics or compact swarms. We present two kinds o asymptotic results in this section: irst, the orm o the swarms or ronts near their singular leading edge and the orm o the swarm near the trailing edges, and then some results or small swarms. Near the leading edge, at x = L, v in (3.) suggests that (x) (L x) /2. Thus substituting appropriate expansions in hal-powers o L x yields, or x < L: (3.6) 2c (L x) /2 4 5 (L x)2 + o( L x 2 ), v 2 2c (L x) 3/2 4 (3.7) 2c (L x) 5/2 + o( L x 5/2 ) 3 5 Near the trailing edge o the swarm, v is nonzero, and the solutions are, locally or x >, 2(v c) x /2 v (3.8) 2 2(v c) x3/2 + o( x 3/2 ),

8 8 PAUL MILEWSKI, XU YANG (3.9) v v + v x + o( x ) Estimates or the speed or small solutions (M, L ) may be obtained by approximating the unction v(x) in terms o a cubic polynomial: (3.2) 3 + 2L v(x) v + v x v L 2 x L + v L 3 x 3, where the coeicients have been ound rom v() = v () = v, v(l) = v (L) =, consistent with the approximations (3.6,3.8) or v. Note that this approximant v is positive but lacks the correct singularity at x = L. In other words, (3.7) is incompatible with (3.2) in the sense that in (3.7), v (L) is unbounded. Using (3.2) in (3.6) or small L, we obtain the results ( v 2c L ) (3.2), 6 (3.22) M c L. Although this approximation is somewhat ad-hoc, it captures remarkably well the the computed values or M/c over the ull range o amplitudes as shown in Figure 3.4. Since we have a one parameter amily o solutions, we can ind L and c as unctions o M alone by considering the expansions or in (3.6,3.8), using (3.2), and calculating M 2 cl 3/2 9 cl 5/2 + O(L 3 ). 3 8 From this relation, together with (3.22) we can obtain equations or c and L as unctions o M (3.23) (3.24) L ( ) /3 9 M / c M 2 6 ( 9 2 ) /3 M 4/3. ( ) 2/3 9 M 2/3, 2 The computed values o c and L together with these approximations are shown in Figure Nonsingular solutions: traveling ronts and waves. Other types o traveling solutions can be obtained i ρ is never zero. Then we cannot set the constant resulting rom the integration o (3.3) to zero. Thus, (3.25) c + v 3 ( 3 ) = A, or, in the phase plane (3.26) (3.27) = v c A 2, v = v.

9 Simple Model or Aggregation numerical asymptotic upper bound lower bound M/c L Fig Computed values o M/c (solid line with circles) or traveling swarms. The upper and lower bounds (3.) are the thick lines, the dashed line is the approximation or small L (3.22). The largest swarm has c.233, M.826, L c L M M /3 Fig Computed values o c and L as unctions o M. The dashed lines are the approximations valid or small M o (3.24). We shall study the various solutions o this system in the (c, A) parameter space. The ixed points satisy v = and thereore (3.28) 2 c A =, ( implying ± = 2 c ± c2 + 4A ). For nonconstant travelling solutions, it is required that A <. This is because or A > there is only one ixed point in the irst quadrant, which rules out homoclinic or heteroclinic orbits. An argument ruling out

10 PAUL MILEWSKI, XU YANG periodic solutions in the case A > is given at the end o the section. Note that since A <, >, (3.28) implies that c >. The nullclines o this system are v = and v = c + A/. The system undergoes a saddle-node biurcation at A = c 2 /4 where the two ixed points are created at = c/2. The Jacobian matrix J o the linearization about ixed points is ( ) A/ 3 / (3.29) J = Its eigenvalues satisy the characteristic polynomial (3.3) λ 2 Tλ + D =. where T = ( + A 3 ) and D = ( + A 3 ) are the trace and determinant o J. Since (3.3) D = (2 c)/ 2 = ± ( ± ) 2 c2 + 4A, + is a node and a saddle. The stability o + depends on the sign o T. + is neutrally stable when T = which implies + n = ( A)/3. Substituting this in the equation (3.28) or +, we obtain (3.32) c = ( A) /3 + ( A) 2/3 as the neutral stability curve in the (c, A) plane. This curve applies to + only or > A > (or A < it is that satisies T = on this curve, which is irrelevant or the phase plane behavior). To decide stability, we calculate (3.33) T A = 3 3A 4 A. From (3.28) we have that / A = (2 c), thereore + / A >, implying that or A <, T A >. Thus the node + is stable or (3.34) 2( A) /2 < c < ( A) /3 + ( A) 2/3, > A >, Region I and becomes unstable or c > ( A) /3 + ( A) 2/3. The node + undergoes a Hop biurcation on the curve (3.32) when a stable limit cycle C s is born. Numerical evidence shows that this limit cycle grows in amplitude and disappears at a homoclinic biurcation or c = c h (A) deined or > A >. Thus, stable limit cycles exist or (3.35) ( A) /3 + ( A) 2/3 < c < c h (A), > A >, Region II and when c = c h there is a homoclinic orbit rom circling + corresponding to a solitary wave o the system. For (3.36) c > c h (A), > A >, or c > 2( A) /2, A < Region III the ixed point + is unstable and there are no limit cycles in the phase plane. A sketch o these regions is given in Figure 3.6. This overall biurcation picture gives rise to many possible traveling waves. In the regime (3.34) there is a heteroclinic orbit with an oscillatory rareaction ront transition rom to + (see Figure 3.7). In the regime (3.35) there are three

11 Simple Model or Aggregation.2.8 A.6 Region I.4 Region II Region III.2 Saddle Node Hop Homoclinic c Fig Parameter space or A < solutions, indicating the biurcations described in the text. The thick solid curve is c = 2( A) /2 above which there are no solutions. The thin solid curve is c = ( A) /3 + ( A) 2/3 along which a limit cycle is born. Along the dashed curve c = c h (A) one has solitary wave solutions on a nonzero background. Note that the curves are only a sketch since in reality the regions labeled are much narrower v θ Fig Phase plane diagram (let) and traveling rareaction ront solutions (right) o the regime I in (3.34). In the phase plane, the dashed lines are the nullclines. For this example c =.8, A =.5. travelling wave solutions: orbits between and C s, + and C s, and the periodic orbit C s (see Figure 3.8). When c = c h (A) there is the solitary homoclinic solution (see Figure 3.9) and one ront-solitary transition solution connecting + to C s. In the regime (3.36) there is a heteroclinic orbit with an oscillatory invasive ront transition rom + to (see Figure 3.). Another interpretation o the ront results arises rom considering solutions tending to the dierent constants +, at ± directly in (3.25) (with the act that there, v ) getting (3.37) c = + + >, A = + <.

12 2 PAUL MILEWSKI, XU YANG v θ Fig Phase plane diagram (let) and constant-to-periodic solutions (right) o the regime II in (3.35). In the phase plane, the dashed lines are the nullclines. Two o the three possible solutions are shown or this set o parameters. A uniorm periodic solution is also possible. For this example c =.5, A = v θ θ Fig Phase plane diagram (let) and solitary (right top) and ront-solitary solutions (right bottom). In the phase plane, the dashed lines are the nullclines. For this example c =.8, A = Thus the ront speed or this problem is the same as the shock speed or the conservation law (3.38) U t + (U 2 ) x =, although the connections between the two states is much more complicated in the present case. We inish this section by inding lower bound on the speed o the periodic waves in terms o their period L and mass (over a period) M and showing that periodic solutions are not permitted or A > (ront solutions are ruled out by the act that there is only one ixed point in this case).

13 Simple Model or Aggregation v v,v θ Fig. 3.. Phase plane diagram (let) and traveling ront solutions (right) o the regime III in (3.36). In the phase plane, the dashed lines are the nullclines. Both and v are shown. For this example c =.5, A =.5. (3.39) Consider, as beore, (3.25) integrated over a period. Then cm + and (3.25) divided by ρ and integrated (3.4) cl + M = A vρ dθ = AL, ρ dθ. We have used that the integral o v is equal to the integral o ρ. Now, (3.4) vρ dθ = ( ) 2 v 2 dθ L v dθ = M2 L L. thereore (3.42) Thereore, or the relevant case, A < (3.43) AL M2 L Mc = M (M Lc) L c M/L. Continuing, we can show that A > is impossible. Using the inequality (3.42) and (3.4) we get (3.44) AL M L A For A >, rom (3.44), we can conclude that ρ dθ. L2 ρ dθ M. However, (3.45) L 2 = ( ) 2 L ρ dθ ρ ρ dθ ρ dθ = M ρ dθ.

14 4 PAUL MILEWSKI, XU YANG ρ y 5 x Fig. 4.. Linear instability o the constant state leading to the ormation o a travelling compact swarm. Initially ρ =. with a small perturbation. Note that m =.63 < M This contradicts the previous inequality unless ρ is constant and thereore we can conclude that no periodic solutions are possible or A >. 4. Time dependent solutions. The time dependent equation, as in the traveling wave case, has a very rich behavior. Here we demonstrate a range o these behaviors and interpret them using the traveling wave solutions we have ound and a linear stability analysis. The time-dependent computations are carried out on a periodic domain with the nonlocal term computed in Fourier space and explicitly in time, and the local, nonlinear, term treated conservatively and implicitly in time. The singular solutions require some care to ensure accurate computations. First, consider a linearization about the constant solution o the orm ρ = ρ + δe σt e ikx we obtain that the perturbation has a growth rate (4.) σ = k2 ρ + k 2 ( ρ ( + k 2 )) ik 2 + k2 + k 2. This means that there is an instability within a band o wavenumbers k (, k ) or ρ <, and that or ρ > there is no instability. The shortest unstable wave has k = ( ρ )/ρ. The maximum growth rate occurs at k max = /ρ and, there, Re σ max = ρ ( ρ ) 2. In Figure 4. we show the evolution rom a constant state with a small perturbation. The solution eventually aggregates into a compact swarm. This behavior o aggregating into one or more compact swarms is universal or suiciently small initial data. I the initial data is larger and yet still unstable, the solution tends to periodic traveling waves. Given a domain o length l, i m is the mass o the initial data over a period and M(L) is the mass o ininite domain swarms o length L, then i m > min(m, M(l)) we can rule out that the solution will tend to a single compact swarm. In Figure 4.2 we show a situation where the initial data is larger than in Figure 4. and the solution tends to a smooth periodic travelling wave. From our numerical experiments, the ultimate state o the system depends on both the initial mass and on the period

15 Simple Model or Aggregation ρ t x Fig Linear instability o the constant state leading to the ormation o a travelling periodic wave. Initially ρ =.2 with a small perturbation. Note that m =.26 > M. Note the similarity o the inal state with the solitary wave solution o Figure Fig Collisions o compact swarms. Snapshots o the collision o 2 swarms o dierent mass. The total mass is.75, less than M. On the let, t = 2, 7, 85. On the right t = 95, 2, 2. Note that: (i) As the aster swarm approaches the smaller one it deorms beore reaching it due to the nonlocal term. (ii) The collision creates a larger swarm (the scale on the right is dierent) and leaves behind some radiation. (iii) The radiation is aggregating (and due to periodicity o the domain the larger swarm has wrapped around). l (especially or l small), leading always to either trains o compact swarms or to periodic waves. The interaction o compact swarms on the ininite line is also interesting. Numerical experiments show that the collision o two swarms tends to orm a single larger swarm and leaves behind some radiation that reorms into a smaller swarm. An example o such collisions is shown in Figure 4.3. Given that constant states with < ρ < are unstable, we can now discuss the stability o the various ront solutions ound in Section 3.3. From (3.28) = when c = A. In Figure 4.4 we show how this line deines regions where both, one or no ixed points are linearly stable. In the region where ± >, which lies in Region III o Figure 3.6, we expect that ront solutions o the type shown in Figure 3., with both states larger than, are stable. The region ± <, we can show, includes

16 6 PAUL MILEWSKI, XU YANG ± > 2 A.5 + >, <.5 ± < c Fig Parameter space or stability o constant states. Region I in Figure 3.6 and, numerically, it appears that it also includes Region II (ie c h (A) < A). Thus we expect solutions o the type shown in Figure 3.7, 3.8 and 3.9 to be unstable. However, periodic travelling solutions appear to be stable and attracting according to numerical calculations. Although these arguments also lead to the conclusion that solitary waves on the real line are unstable, periodic solutions o long period and large amplitude, which are very close to the solitary wave solution, seem to be stable (see, or example Figure 4.2). This phenomenon appears to be due to the act that the periodic soliton sweeps through the domain quickly, destroying any growing instability which travels more slowly and, which, since the background density is low, has small growth rate. When + >, <, which is also in Region III o Figure 3.6, the solutions o the type shown in Figure 3. are also possible. Ahead o the ront, < and thereore we can expect instability i the perturbations extend to +. However, i perturbations are restricted to be localized near the ront, we conjecture that the ront is stable. For singular ronts (see section 3.), i the state behind the ront has ρ <, we expect it to be unstable. Examples o the propagation o singular ronts and their (in)stability are shown in igure Bi-directional model. One shortcoming o the model we present is that organisms are always moving in the same direction. A more realistic model is when organisms can reverse their direction o sensing and motion. We thereore consider two populations o right- and let-sensing organisms. Using the densities ρ +, and ρ respectively, deining ρ ρ + + ρ, introducing the sensing kernels: K + (x) = K(x), K = K( x) and modeling a turning process, we can write, or example, (5.) (5.2) ρ + t + [ ρ + (K + ρ ρρ x ) ] x = α(ρ+ ρ ) + β ρ + ρ (ρ + ρ ), ρ t + [ ρ (K ρ ρρ x ) ] x = α(ρ ρ + ) + β ρ + ρ (ρ ρ + ). The term proportional to α corresponds to a Poisson process or organisms turning around and the β term corresponds to organisms turning when there is a large density moving in the opposite direction. Note that in the lux, we have chosen the speed to depend on the ull population, and that the system preserves the total number o

17 Simple Model or Aggregation Fig Two examples o singular ront propagation. The initial data (dotted curve) is a nonzero constant state or x < 2π and zero or x > 2π with a random perturbation or < x < 2π. Let: Solutions at various times when the state behind the ront has ρ > and the dynamics leads to a stable ront. The solution is shown at t =,,2, 3, 4, 5. Right: Solutions when the state behind the ront is unstable (ρ < ) and the ront breaks up into a combination compact swarms and nonsingular oscillations. The solution is shown at t = and t = Fig. 5.. Spreading o localized initial data in a bi-directional model with a poisson process or turning. Here, α = 2 and β =. organisms. For α =, and i one o the ρ is initially zero, then it remains zero and the dynamics reduces to the one-way model we have discussed. In Figure 5. we show an example o a calculation with β = where a localized mixed population spreads. 6. Conclusion. We have introduced a simple continuum swarming model with directional sensing which leads to the aggregation o organisms into compact traveling swarms, when local densities are suiciently small. At larger organism densities one observes traveling waves. Between regions o dierent organism densities there are ront solutions. The model is simple enough that one can make substantial progress at understanding the solutions analytically. REFERENCES

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