The biodiversity of food webs is composed of horizontal (i.e. within trophic levels) and vertical diversity (i.e. the number of trophic levels). Understanding their joint effect on stability is a key challenge. Theory mostly considers their individual effects and focuses on small perturbations near equilibrium in hypothetical food webs. Here, we study the joint effects of horizontal and vertical diversity on the stability of hypothetical (modelled) and empirical food webs. In modelled food webs, horizontal and vertical diversity increased and decreased stability, respectively, with a stronger positive effect of producer diversity on stability at higher consumer diversity. Experiments with an empirical plankton food web, where we manipulated horizontal and vertical diversity and measured stability from species interactions and from resilience against large perturbations, confirmed these predictions. Taken together, our findings highlight the need to conserve horizontal biodiversity at different trophic levels to ensure stability.

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In the average method of modified small parameters, the synchronization of two coupled exciters is converted to a problem on the existence and stability of zero solutions for the average differential equations of small parameters over the average period of two exciters. To implement frequency capture, the torque of frequency capture should be greater than or equal to the absolute value of the difference between the residual electromagnetic torques of the two motors. Because each exciter is involved in the motion of the vibrating system it has excited, its relative moment of inertia is reduced. The reduction is proportional to half its coefficient of cosine effect of phase angles (CCEPA). Because one of the exciters is involved in the motion excited by the other, a coupled moment of inertia exists for the two exciters. The stability of the synchronization of the two exciters is affected by the reduction of their relative moments of inertia and their moment of coupling inertia. For the synchronization to be stable, two conditions must be satisfied: (1) the non-dimensional relative moments of inertia of the two exciters are all greater than zero, and (2) four times the product of their non-dimensional relative moments is greater than the square of the coefficient of coupled cosine effect (CCCPA). The stability of synchronization depends solely on the ratios of the masses of the two exciters to the mass of the vibrating system and the ratio of the distance between one exciter and the centroid of the rigid frame to the equivalent rotating radius of the vibrating system about its centroid of the rigid frame, and is independent of the parameters of the two induction motors.

Aquatic microcosm studies often increase either chamber height or base diameter (to increase water volume) to test spatial ecology theories such as “scale” effects on ecological processes, but it is unclear whether the increase of chamber height or base diameter have the same effect on the processes, i.e., whether the effect of the shape of three-dimensional spaces is significant. We orthogonally manipulated chamber height and base diameter and determined swimming activity, average swimming velocity and grazing rates of the cladocerans Daphnia magna and Moina micrura (on two algae Scenedesmus quadricauda and Chlorella vulgaris; leading to four aquatic algae-cladoceran systems in total) under different microcosm conditions. Across all the four aquatic systems, increasing chamber height at a given base diameter significantly decreased the duration and velocity of horizontal swimming, and it tended to increase the duration but decrease the velocity of vertical swimming. These collectively led to decreases in both average swimming velocity and grazing rate of the cladocerans in the tall chambers (at a given base diameter), in accordance with the positive relationship between average swimming velocity and grazing rate. In contrast, an increase of base diameter at a given chamber height showed contrasting effects on the above parameters. Consistently, at a given chamber volume increasing ratio of chamber height to base diameter decreased the average swimming velocity and grazing rate across all the aquatic systems. In general, increasing chamber depth and base diameter may exert contrasting effects on zooplankton behavior and thus phytoplankton-zooplankton interactions. We suggest that spatial shape plays an important role in determining ecological process and thus should be considered in a theoretical framework of spatial ecology and also the physical setting of aquatic microcosm experiments.

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