
MBW:Locust Phase Change and SwarmingFrom MathBioContentsExecutive SummaryIn Locust Dynamics: Behavioral Phase Change and Swarming^{[1]}, Topaz et. al discuss necessary conditions for an outbreak of marching hopper bands of locusts. Outbreaks are triggered by a variety of factors, such as visual, olfactory, and tactile cues. Typically, locusts lead solitary lifestyles in arid regions. When overcrowding occurs, the locusts tend to change their behavior in the newly formed group to be gregarious, rather than solitarious. Topaz et. al developed a partial integrodifferential equation model in order to understand the dynamics of hopper band formation at the population level. The model is based on reactiondiffusion equations that relate the locusts’ behavioral phase changes and spatial movement. Stability analysis identifies conditions for an outbreak. Topaz et. al proceed to perform a model reduction in order to quantify the temporal dynamics, the subset of the population that will eventually group together, and the time until a swarm occurs. Finally, a numerical simulation reveals the structure of the swarm. They ultimately offer suggestions for locust control strategies:
Context/Biological phenomenon under considerationTo see a locust swarm in action, check out this video.
The depleting resources in the area forces them to migrate en masse in search of new feeding grounds. This swarm of locusts can consist of millions of individuals, and it poses a huge threat to farmers. Swarms can travel a few hundred kilometers per day and span up to a thousand square kilometers ^{[1]}. Swarms are notorious for leaving farms without a single plant growing, and billions of dollars in crops are lost. HistoryPrevious locust models are based on Lagrangian simulations, where the position, velocity, and interactions of individual locusts are tracked. Also, the focal point of many current models is interactions with clusters of resources and varying environmental conditions. Furthermore, some models use anisotropic interactions (different responses to anterior or posterior neighbors) or consider Newtonian dynamics^{[1]}. Topaz et. al present a model that is based on population density (Eulerian), rather than individuals. This allows them to use techniques of partial differential equations and their extensions (integroPDEs). They consider the attractiverepulsive social interactions between locusts in their model instead of interactions between resource clusters and changing environmental conditions. Lastly, the authors rely on isotropic interactions (as opposed to anisotropic ones) in their model. Assumptions
Model constructionClassic swarm modeling involves a conserved population density field travelling at a velocity that arises from social interactions:
The contribution of of a small group of individuals at location to the force on the individual at position is given by:
Therefore,
The reader should note that the term is radially symmetric. Let us use some notation to represent this integral:
This is a nonlocal interaction model, and its purpose is to capture spatially distributed interactions, as opposed to pure PDEs (which describe interactions over infinitesimal ranges). This basic model is further adapted for biphasic insects below (the Full model). Parameters
Full modelAssumptions
Model constructionThere are separate density fields for the solitarious and gregarious phases , and these phases sum to the total density:
The reactiondiffusion equations of the model are:
Where the velocities are:
We now require equations for the solitarious/gregarious social interactions and the two densitydependent conversion rate functions, , which represent the rate of gregarious solitarious and the rate of solitarious gregarious, respectively:
Failed to parse (lexing error): Q_g(\mathbf{ xx'}) = R_g e^{\mathbf{xx'}/r_g} – A_g e^{\mathbf{xx'}/a_g}
Conversion rates from one phase to the otherBy biological observation, we know that the time to gregarization is relatively fast compared to the time to solitarization. Thus, the equations for are justified. It should be noted that decreases with , while increases with to the saturation point, . Critical distanceAs indicated above, it should be noted that is a decaying exponential function for all choices of and , so it is purely repulsive. However, , being the difference of two exponentials, suggests that there is an equilibrium, where there is no net contribution to the velocity. The equilibrium point is found by solving .
Failed to parse (lexing error): R_ga_g – A_gr_g > 0
Failed to parse (lexing error): A_ga_g^2 – R_gr_g^2 > 0
These conditions are based on existing literature that shows cohesiveness requires parameters in must be such that locusts are led to clumping ^{[1]}. ParametersThe numerical values of these parameters are based on estimates from biological literature ^{[1]}.
Analysis/interpretationTopaz et. al fist find the homogeneous steady state solution to the reactiondiffusion equations and analyze the solutions using linear stability analysis, which is equivalent to finding the eigenvalues of the linearized system. The stability analysis reveals the condition for the onset of a swarm. The authors then proceed to perform numerical simulations in one spatial dimension to identify the structure of the swarm. They find the model exhibits hysteresis on the population level (macroscopic properties of the reactiondiffusion equations, which are outputs of the model). Homogeneous steady states (HSS)The steady state solutions to the reactiondiffusion equations for low density are:
For large density :
The middle curve in each set of curves is the solution for the default choice of parameters. The other curves show parameter sensitivity. In the gray and red regions, the HSS is linearly unstable to small perturbations. At the critical density , reaches a maximum, and continues to grow. As a result of the authors’ choice of the default parameters, occurs at , but this is not always the case, as is seen in the top and bottom curves (the 25th and 75th percentile values). The general case where is expressed below, where the sum of the fractions of solitarious and gregarious locusts equals one:
The gregarious state necessarily becomes the dominant one as density increases:
Linear stability analysisLinear stability results depend on Fourier transforms of the interaction potentials . The eigenvalue determines the stability of the locust model:
Based on the equations above, Topaz et. al found the instability condition in terms of : Failed to parse (lexing error): \phi_g > \phi_g^*=\frac{R_sr_s^2}{R_sr_s^2 – R_gr_g^2 +A_ga_g^2 }
vary along the horizontal and vertical axes. The contours are critical values of , the rescaled density. For , the solutions are unstable. The arrow along the horizontal axis shows where moves as the rate of gregarization increases. The arrow along the vertical axis shows where travels as the density threshold for gregarization decreases. It should be noted that one should use for an accurate biological interpretation, not (). The default parameters yield the following results:
The authors found extremely disparate results with different parameters, : Using the default parameters (blue and dashed green curves of plots A, B), they found the wave number corresponding to the most rapidly growing perturbation. Their findings are shown in the plot below: Again, the middle curve of this plot matches the middle curve in plots A,B. For the middle curve, saturation occurs at . This value indicates fastgrowing perturbations occur on the length scale of a few locust bodies because the length scale . Unfortunately, this linear stability analysis only applies to the case where we observe small perturbations of uniform steady states. It does not predict longterm or largeamplitude dynamics, and neither does it accurately describe large perturbations. Furthermore, using this model to analyze small perturbations in a nonuniform steady state would also be erroneous. Numerical simulationTopaz et. al construct a one dimensional numerical simulation to examine the structure of the swarm. The parameters of the simulation are as follows:
The results are shown in the figures below, where the first one is with the default set of parameters and the second one is with the alternative set of parameters: This snapshots of the number of solitarious and gregarious locusts at different times illustrates the transition from a population of initially solituarious locusts () to a gregarious swarm (). HysteresisThe numerical results suggest the model has populationlevel hysteresis for the alternative parameters (macroscopic properties of the reactiondiffusion equations, which are outputs of the model): Unstable behavior is represented by dashed lines and stable behavior is represented by the solid lines. The red curve is the HSS solution, where there is no clustering. The green curve represents the clustered state from the numerical simulation. The asterisk on the horizontal axis represents the point of linear instability, where stability jumps from the red curve to the green curve. As increases, the stable solution is along the red curve. As passes through the asterisk, stability changes to the green curve. As is decreased from this point, stability remains on the green curve (even for small values of ). Hysteresis suggests that once a locust swarm has formed, it cannot be eliminated by reducing the density of the swarm. DiscussionThe results of this investigation can be summarized as follows:
Topaz et. al suggest the following locust management strategies:
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