N. A. Hill

Growth of bioconvection patterns in a suspension of gyrotactic micro-organisms in a layer of finite depth

The effect of gyrotaxis on the linear stability of a suspension of swimming, negatively buoyant micro-organisms is examined for a layer of finite depth. In the steady basic state there is no bulk fluid motion, and the upwards swimming of the cells is balanced by diffusion resulting from randomness in their shape, orientation and swimming behaviour. This leads to a bulk density stratification with denser fluid on top. The theory is based on the continuum model of Pedley, Hill & Kessler (1988), and employs both asymptotic and numerical analysis. The suspension is characterized by five dimensionless parameters : a Rayleigh number, a Schmidt number, a layer-depth parameter, a gyrotaxis number G, and a geometrical parameter measuring the ellipticity of the micro-organisms. For small values of G, the most unstable mode has a vanishing wavenumber, but for sufficiently large values of G, the predicted initial wavelength is finite, in agreement with experiments. The suspension becomes less stable as the layer depth is increased. Indeed, if the layer is sufficiently deep an initially homogeneous suspension is unstable, and the equilibrium state does not form. The theory of Pedley, Hill & Kessler (1988) for infinite depth is shown to be appropriate in that case. An unusual feature of the model is the existence of overstable or oscillatory modes which are driven by the gyrotactic response of the micro-organisms to the shear at the rigid boundaries of the layer. These modes occur at parameter values which could be realized in experiments.

Development and stability of gyrotactic plumes in bioconvection

Using the continuum model of Pedley, Hill & Kessler (1988) for bioconvection in a suspension of swimming, gyrotactic micro-organisms, we investigate the existence and stability of a two-dimensional plume in tall, narrow chambers with stress-free sidewalls. The system is governed by the Navier{Stokes equations for an incompressible fluid coupled with a micro-organism conservation equation. These equations are solved numerically using a conservative nite-dierence scheme. In suciently deep chambers, the plume is always unstable to both varicose and meandering modes. A linear stability analysis for an innitely long plume predicts the growth rates of these instabilities, explains the mechanisms, and is in good agreement with the numerical results.

Periodic arrays of gyrotactic plumes in bioconvection

Using the continuum model of Pedley et al. [J. Fluid Mech. 195, 223 (1988)] for bioconvection in a suspension of swimming, gyrotactic micro-organisms, the existence and stability of periodic arrays of two-dimensional plumes in deep chambers are investigated. The system is governed by the Navier–Stokes equations for an incompressible fluid coupled with a micro-organism conservation equation. These equations are solved numerically using a conservative finite-difference scheme. In sufficiently deep chambers, the plumes are sometimes unstable to varicose or meandering modes. A linear stability analysis for an infinitely deep plume predicts the growth rates of these instabilities and agrees well with the numerical results.

Linear bioconvection in a suspension of randomly swimming, gyrotactic micro-organisms

We have analyzed the initiation of pattern formation in a layer of finite depth for Pedley and Kessler’s new model [J. Fluid Mech. 212, 155 (1990)] of bioconvection. This is the first analysis of bioconvection in a realistic geometry using a model that deals with random swimming in a rational manner. We have considered the effects of a distribution of swimming speeds, which has not previously received attention in theoretical papers and find that it is important in calculating the diffusivity. Our predictions of initial pattern wavelengths are reasonably close to the observed ones but better experimental measurements of key parameters are needed for a proper comparison.

ORIENTATION OF SWIMMING FLAGELLATES BY SIMULTANEOUSLY ACTING EXTERNAL FACTORS1

The directionality of phototaxis combined with gravitaxis was investigated experimentally for populations of the swimming alga Euglena gracilis Klebs. Two irradiances were used: a “weak” irradiance to elicit positive phototaxis and a “strong” irradiance to elicit negative phototaxis. In addition, by changing the density of cells in the suspension, the number of collisions between cells was varied to determine the effects of these collisions on the distribution of swimming directions in both the absence and the presence of illumination. We found that positive phototaxis was associated with a broader distribution of swimming directions than was negative phototaxis. In the latter case, the effect of phototaxis dominated over that of gravitaxis. Experiments on another swimming alga, Chlamydomonas nivalis Wille, showed that collisions between cells degraded the directionality of gravitaxis.

Taylor dispersion of gyrotactic swimming micro-organisms in a linear flow

The theory of generalized Taylor dispersion for suspensions of Brownian particles is developed to study the dispersion of gyrotactic swimming micro-organisms in a linear shear flow. Such creatures are bottom-heavy and experience a gravitational torque which acts to right them when they are tipped away from the vertical. They also suffer a net viscous torque in the presence of a local vorticity field. The orientation of the cells is intrinsically random but the balance of the two torques results in a bias toward a preferred swimming direction. The micro-organisms are sufficiently large that Brownian motion is negligible but their random swimming across streamlines results in a mean velocity together with diffusion. As an example, we consider the case of vertical shear flow and calculate the diffusion coefficients for a suspension of the alga Chlamydomonas nivalis. This rational derivation is compared with earlier approximations for the diffusivity.

Bioconvection in a suspension of phototactic algae

In this paper we develop a new generic model for phototaxis in a suspension of microscopic swimming algae. Phototaxis is a directed swimming response dependent upon the light conditions sensed by the microorganisms. Positive phototaxis consists of motions directed toward the source of illumination and negative phototaxis is motion directed away from it. The model also incorporates the effect of shading whereby microorganisms nearer the light source absorb and scatter the light before it reaches those further away. This model of phototaxis and shading is then used to analyse the linear stability of a suspension of phototactic algae, uniformly illuminated from above, that swim in a fluid which is slightly less dense then they are. In the basic state there are no fluid motions and the up and down swimming caused by positive and negative phototaxis is balanced by diffusion, with the result that a horizontal, concentrated layer of algae forms, the vertical position of which depends on the light intensity. This leads to a bulk density stratification with a gravitationally stable layer of fluid overlying an unstable layer. When the resulting density gradient becomes large enough, a Rayleigh–Taylor type instability initiates fluid motions in the lower, unstable region that subsequently penetrate the upper, locally stable region. The behaviour of the suspension is characterized by four parameters: a layer depth parameter d , a Rayleigh number R , a Schmidt number S c , and a sublayer position parameter C that specifies the vertical position of the sublayer in the fluid. Linear stability analysis of the basic state indicates that initial pattern wavelengths and the critical value of R at which the suspension becomes unstable are affected by the position of the sublayer, which is determined by the intensity of the light source, and by the type of boundary at the top of the fluid. Oscillatory modes of disturbance are also predicted for certain parameter ranges, driven by cells swimming vigorously upwards towards darker, concentrated downwelling regions and thus creating lower, positively buoyant regions which oppose the fluid motion in the convection patterns.

A multiscale maximum entropy moment closure for locally regulated space-time point process models of population dynamics

The prevalence of structure in biological populations challenges fundamental assumptions at the heart of continuum models of population dynamics based only on mean densities (local or global). Individual-based models (IBMs) were introduced during the last decade in an attempt to overcome this limitation by following explicitly each individual in the population. Although the IBM approach has been quite useful, the capability to follow each individual usually comes at the expense of analytical tractability, which limits the generality of the statements that can be made. For the specific case of spatial structure in populations of sessile (and identical) organisms, space–time point processes with local regulation seem to cover the middle ground between analytical tractability and a higher degree of biological realism. This approach has shown that simplified representations of fecundity, local dispersal and density-dependent mortality weighted by the local competitive environment are sufficient to generate spatial patterns that mimic field observations. Continuum approximations of these stochastic processes try to distill their fundamental properties, and they keep track of not only mean densities, but also higher order spatial correlations. However, due to the non–linearities involved they result in infinite hierarchies of moment equations. This leads to the problem of finding a ‘moment closure’; that is, an appropriate order of (lower order) truncation, together with a method of expressing the highest order density not explicitly modelled in the truncated hierarchy in terms of the lower order densities. We use the principle of constrained maximum entropy to derive a closure relationship for truncation at second order using normalisation and the product densities of first and second orders as constraints, and apply it to one such hierarchy. The resulting ‘maxent’ closure is similar to the Kirkwood superposition approximation, or ‘power-3’ closure, but it is complemented with previously unknown correction terms that depend mainly on the avoidance function of an associated Poisson point process over the region for which third order correlations are irreducible. This domain of irreducible triplet correlations is found from an integral equation associated with the normalisation constraint. This also serves the purpose of a validation check, since a single, non-trivial domain can only be found if the assumptions of the closure are consistent with the predictions of the hierarchy. Comparisons between simulations of the point process, alternative heuristic closures, and the maxent closure show significant improvements in the ability of the truncated hierarchy to predict equilibrium values for mildly aggregated spatial patterns. However, the maxent closure performs comparatively poorly in segregated ones. Although the closure is applied in the context of point processes, the method does not require fixed locations to be valid, and can in principle be applied to problems where the particles move, provided that their correlation functions are stationary in space and time.

Reframing the concept of alternative livelihoods.

Abstract Alternative livelihood project (ALP) is a widely used term for interventions that aim to reduce the prevalence of activities deemed to be environmentally damaging by substituting them with lower impact livelihood activities that provide at least equivalent benefits. ALPs are widely implemented in conservation, but in 2012, an International Union for Conservation of Nature resolution called for a critical review of such projects based on concern that their effectiveness was unproven. We focused on the conceptual design of ALPs by considering their underlying assumptions. We placed ALPs within a broad category of livelihood‐focused interventions to better understand their role in conservation and their intended impacts. We dissected 3 flawed assumptions about ALPs based on the notions of substitution, the homogenous community, and impact scalability. Interventions based on flawed assumptions about peoples needs, aspirations, and the factors that influence livelihood choice are unlikely to achieve conservation objectives. We therefore recommend use of a sustainable livelihoods approach to understand the role and function of environmentally damaging behaviors within livelihood strategies; differentiate between households in a community that have the greatest environmental impact and those most vulnerable to resource access restrictions to improve intervention targeting; and learn more about the social–ecological system within which household livelihood strategies are embedded. Rather than using livelihood‐focused interventions as a direct behavior‐change tool, it may be more appropriate to focus on either enhancing the existing livelihood strategies of those most vulnerable to conservation‐imposed resource access restrictions or on use of livelihood‐focused interventions that establish a clear link to conservation as a means of building good community relations. However, we recommend that the term ALP be replaced by the broader term livelihood‐focused intervention. This avoids the implicit assumption that alternatives can fully substitute for natural resource‐based livelihood activities.

Hydrodynamic diffusion of a sphere sedimenting through a dilute suspension of neutrally buoyant spheres

The motion of a heavy sphere sedimenting through a dilute background suspension of neutrally buoyant spheres is analysed for small Reynolds number and large PBclet number. For this particular problem, it is possible not only to calculate the mean velocity of the heavy particle, but also the variance of the velocity and the coefficient of hydrodynamic diffusivity . Pairwise, hydrodynamic interactions between the heavy sphere and the background sphere are considered exactly using volume integrals and a trajectory analysis. Explicit formulae are given for the two limiting cases when the radius of the heavy sphere is much greater and much less than that of the background spheres, and numerical results are given for moderate size ratios. The mean velocity is relatively insensitive to the ratio of the radius of the background spheres to that of the heavy sphere, unless this ratio is very large, whereas the hydrodynamic diffusivity increases rapidly as the radius ratio is increased. The predictions are in reasonable agreement with the results of falling-ball rheome try experiments.