MORTAL CREEPERS SEARCHING FOR A TARGET

MORTAL CREEPERS SEARCHING FOR A TARGET D. Campos, E. Abad, V. Méndez, S. B. Yuste, K. Lindenberg Barcelona, Extremadura, San Diego We present a simple paradigm for detection of an immobile target by a Lévy walker (or creeper) with a finite lifetime. The motion of the walker is characterized by linear displacements at a fixed speed and exponentially distributed duration, interrupted by random changes in the direction of motion and resumption of motion in the new direction with the same speed. A walker with these characteristics is also called a "creeper." The walker may die at any time according to an exponential decay law characterized by a finite mean death rate. We consider the efficiency of the target search process, characterized by the probability that the walker will eventually detect the target. We find that there is an optimal frequency that depends on the lifetime and frequency of change of direction that maximizes the probability of eventual target detection. We also consider the survival probability of the target in the presence of many independent walkers, related to the well-studied standard target problem" in which many walkers start at random locations at the same time. MINECO, Generalitat de Catalunya, ONR

A CREEPER is a Lévy walker with a constant speed and an exponential distribution of direction changing times A MORTAL CREEPER can die as it walks, with an exponential distribution of death times We will start with motion in d = System size L = Target of radius R (= in d) Creeper, point particle Speed v

A CREEPER is a Lévy walker with a constant speed and an exponential distribution of direction changing times A MORTAL CREEPER can die as it walks, with an exponential distribution of death times Our TARGET is fixed, does not move. TARGET and CREEPER die when the CREEPER hits the TARGET First a SINGLE CREEPER. Later a bit about MANY CREEPERS Mainly d=. Later perhaps a bit about d=2

OUR QUESTION: What is the best trajectory to maximize the probability that the creeper kills the target? Ballistic? Turns? How many turns? NOTE: All results in d= are analytic! Note also: this work is (closely?) related to that of the following two references:. Intermittent random walks for an optimal search strategy: onedimensional case, J. Phys. Condens. Matter 9 (27) 6542. G Oshanin, H S Wio, KL, and S F Burlatsky. 2. Efficient search by optimized intermittent random walks, J. Phys. A: Math. Theor. 42 (29) 4348. G Oshanin, KL, H S Wio, and S Burlatsky.

Master equation: first an immortal creeper, no target j(x, t x )dxdt = probability that the creeper completes a displacement at [x, x + dx] and thus turns during [t, t + dt] j (x) = (x x ) (t) (x, t) = probability density of having gone a distance x in time interval t between consecutive turns p(x, t x ) = probability that the creeper is in [x, x + dx] at time t given that it was at x at t = (x, t)dx / (x ± vt) = distance covered in time t before next turn

Master equation: first an immortal creeper, no target j(x, t x )dxdt = probability that the creeper completes a displacement at [x, x + dx] and thus turns during [t, t + dt] j (x) = (x x ) (t) (x, t) = probability density of having gone a distance x in time interval t between consecutive turns ************************************ j(x, t x )= Z t dt Z dx j(x x,t t x ) (x,t )+j

Master equation: first an immortal creeper, no target j(x, t x )dxdt = probability that the creeper completes a displacement at [x, x + dx] and thus turns during [t, t + dt] p(x, t x ) = probability that the creeper is in [x, x + dx] at time t given that it was at x at t = (x, t)dx / (x ± vt) = distance covered in time t before next turn ************************************ p(x, t x )= Z t dt Z dx j(x x,t t x ) (x,t )

j(x, t x )= Z t dt Z dx j(x x,t t x ) (x,t )+j p(x, t x )= Z t dt Z dx j(x x,t t x ) (x,t ) Exponentially decaying waiting time between changes in direction (turns) (x, t) = 2 [ (x + vt)+ (x vt)]! exp(!t) (x, t) = (x, t)!! = Mean duration of motion in one direction

More probabilities, rates: q(x, t x ) = rate at which a creeper that starts at x steps on x f(x, t x ) = rate of first step on (encounter with) x q(x, t x )=f(x, t x )+ Z t q(x, t x, t )f(x, t x )dt Next: Introduce a target at x = x t

Next: Introduce a target at x = x t Survival probability of target S(t) = Z t f(x t,t x )dt ASYMPTOTIC SURVIVAL PROBABILITY Z S = f(x t,t x )= In dimension d = target is certain to die

Now add creeper mortality: Exponentially distributed times to death '(t) = probability that creeper is still alive at time t '() =! m = mean lifetime of creeper p (x, t x )='(t)p(x, t x ) q (x, t x, t )=['(t)/'(t )]q(x, t x, t ) f (x, t x )=['(t)/'()]f(x, t x )='(t)f(x, t x )

q (x, t x )=f (x, t x )+ Z t q (x, t x, t )f (x, t x )dt S (t) = Z t f (x t,t x )dt ASYMPTOTIC SURVIVAL PROBABILITY S = Z f (x t,t x )dt >

time SPLAT!!! or DEAD!!! One creeper, one target Speed v= Turning distribution: exponential Mean:! Mean Death time distribution: exponential Mean:! m

NOTE: All results in d= are analytic! OUR QUESTION: What is the best trajectory to maximize the probability that the creeper kills the target? Ballistic? Turns? How many turns?

OUR QUESTION: What is the best trajectory to maximize the probability that the creeper kills the target? Ballistic? Turns? How many turns? Principal parameters: Maximize Turning rate Dying rate!! m S Minimize S

xt x =. (circles) x t x =. circles x t x = 5 triangles S! m =.2! m =2 4! m =2 3......! S = p!m (! m +!)! m + p! m (! m +!) exp p!m (! m +!) x t x Value of! =! opt is tradeoff between not losing patience in case target is close, but breaking persistence in case target is far.

xt x =. (circles) x t x =. circles x t x = 5 triangles S! m =.2! m =2 4! m =2 3....... If creeper starts closer to the target it has a better chance of getting there. 2. If creeper lives for a long time it has a better chance of arriving at the target! 3. Ballistic motion is sometimes (often?) optimal. (! = ) Value of! =! opt! m = is tradeoff between not losing patience in case target is close, but breaking persistence in case target is far. (! opt x t x ) x t x (2! opt x t x )

xt x =. (circles) L = x t x =. circles x t x = 5 triangles! m =.2 S! m =2 4! m =2 3......! L = 2 Difficult to see differences, but there are some

xt x =. (circles) L = x t x =. circles x t x = 5 triangles! m =.2 S L = 2! m =2 4! m =2 3......!! opt L = CIRCLES - Ballistic motion always optimal L = SQUARES L = DIAMONDS} Ballistic motion L! optimal at low and at high mortality rates TRIANGLES - Ballistic motion optimal at high mortality rates! m Note coincidence of curves at high mortality rate

ωopt..8.6.4.2 7 6 5 4 3 2 ω m L = L = L = L! CIRCLES SQUARES DIAMONDS TRIANGLES Schematic of trajectories of the three regimes shown above for intermediate values of L

Span of mortal walker as a function of frequency of reorientation span = 2! m + p! m (! m +!)

ωopt.35.3.25.2.5..5 5 4 3 2 ω m 2d results only numerical LxL square lattices, PBC Behavior similar to d p (xt x ) 2 +(y t y ) 2 =2 L = 25 circles L = squares L = 4 diamonds L! triangles

SUMMARY AND CONCLUSIONS, all results asymptotic all d results are ANALYTIC. One creeper that does not die in an infinite system will eventually find the target in d= and d=2 with probability find the target with probability < in d = 3 or greater will find the target with probability in any dimension if system is finite 2. When L is infinite, in most cases (but not all) the best strategy for a creeper to find the target is to move ballistically,! opt =. For a few parameter combinations! opt 6=. 3. For a finite density of creepers initially distributed at random (the target problem ) the best strategy for target detection is ballistic motion. 4. A comparison table for best strategies for target detection For finite L in d= Small L: ballistic motion Intermediate L:! opt 6=. Large L: ballistic motion! opt =.! opt =. For infinite L in d= Small! m : ballistic motion! opt =.! m! opt 6=. Intermediate : Large! m : ballistic motion! opt =.

SOME QUESTIONS FOR CONTINUED STUDY. Behavior in 2d similar to that in d. How about 3d or more? 2. Waiting time distribution between turns long-tailed (Lévy)? 3. Long-tailed distribution of times to death? 4. Moving target? 5. Limited search time? 6. Distribution of velocities