Evaluating the Structure of Enemy Biodiversity Effects on Prey Informs Pest Management

Table 1

Best model structure and main parameter estimates relative to enemy biodiversity effects on whitefly nymph survival.

Code	Model structure	Model selection diagnostics					Maximum likelihood estimates of model parameters
		ΔQAICc	𝑤𝑖	𝐾𝑖	𝑐𝑖	𝑅2𝑖	𝑘NB	𝑝𝐶Ny	𝑝^′𝑀𝑚	𝑝^′𝑀𝑚+𝑁𝑡	𝑝^′𝑀𝑚+𝑂𝑥	𝑝^′𝑀𝑚+𝐿𝑖	𝑝^′𝑁𝑡	𝑝^′𝑁𝑡+𝑂𝑥	𝑝^′𝑁𝑡+𝐿𝑖	𝑝^′𝑁𝑡+𝑀𝑚
S1	𝑝(Ny)(𝐶,𝑖,𝐵𝑖,𝑎𝑖,𝑀); all	57.6253	0.0000	27	2.3675	1.0000	0.0000	0.0729	0.3004	0.0000	0.0000	0.4914	0.1611	0.0256	0.1192	0.2908
S2	𝑝(Ny)(𝐶,𝑖,𝐵𝑖,𝑎𝑖,𝑀);𝑝^′𝑂𝑥=0	34.4510	0.0000	21	2.2286	0.9944	0.0005	0.0869	0.2860	0.4184	0.2474	0.4899	0.1097	0.1541	0.1787	0.1552
S3	𝑝(Ny)(𝐶,𝑖,𝐵𝑖,𝑎𝑖,𝑀);𝑝^′𝑂𝑥=𝑝^′𝐿𝑖=0	14.2799	0.0004	15	1.8687	0.9810	0.0015	0.0894	0.3128	0.3905	0.2870	0.5029	0.1163	0.1633	0.1802	0.1700
S4	𝑝(Ny)(𝐶,𝑖,𝐵,𝑎𝑖,𝑀);𝑝^′𝑂𝑥=𝑝^′𝐿𝑖=0	11.5475	0.0014	14	1.8225	0.9787	0.0017	0.0914	0.3001	0.3798	0.2615	0.4784	0.1148	0.1614	0.1785	0.1678
S5	𝑝(Ny)(𝐶,𝑖,𝐵,𝑎,𝑀);𝑝^′𝑂𝑥=𝑝^′𝐿𝑖=0	26.3280	0.0000	13	1.6959	0.9072	0.0075	0.0906	0.1021	0.1645	0.0850	0.1304	0.1336	0.2037	0.2306	0.1646
S6	𝑝(Ny)(𝑖,𝐵,𝑎𝑖,𝑀);𝑝^′𝑂𝑥=𝑝^′𝐿𝑖=0	24.1824	0.0000	13	1.4922	0.8872	0.0092	—	0.2716	0.2591	0.2200	0.3759	0.1208	0.1598	0.1878	0.1662
S7	𝑝(Ny)(𝐶,𝑖,𝐵,𝑎𝑖,𝑀);𝑝^′𝑂𝑥=𝑝^′𝐿𝑖=0;𝑝^′𝑀𝑚=𝑝^′𝑀𝑚+𝑁𝑡	8.3366	0.0070	13	1.7840	0.9783	0.0018	0.0914	0.3019	=𝑝^′𝑀𝑚	0.2616	0.4788	0.1159	0.1633	0.1804	0.1739
S8	𝑝(Ny)(𝐶,𝑖,𝐵,𝑎𝑖,𝑀);𝑝^′𝑂𝑥=𝑝^′𝐿𝑖=0;𝑝^′𝑀𝑚=𝑝^′𝑀𝑚+𝑁𝑡=𝑝^′𝑀𝑚+𝑂𝑥	5.3157	0.0317	12	1.7565	0.9785	0.0017	0.0917	0.2976	=𝑝^′𝑀𝑚	=𝑝^′𝑀𝑚	0.5006	0.1162	0.1638	0.1809	0.1757
S9	𝑝(Ny)(𝐶,𝑖,𝐵,𝑎𝑖,𝑀);𝑝^′𝑂𝑥=𝑝^′𝐿𝑖=0;𝑝^′𝑀𝑚=𝑝^′𝑀𝑚+𝑃𝑗	5.1225	0.0350	11	1.7795	0.9759	0.0020	0.0917	0.3323	=𝑝^′𝑀𝑚	=𝑝^′𝑀𝑚	=𝑝^′𝑀𝑚	0.1152	0.1619	0.1792	0.1704
S10	𝑝(Ny)(𝐶,𝑖,𝐵,𝑎𝑖,𝑀);𝑝^′𝑂𝑥=𝑝^′𝐿𝑖=0;𝑝^′𝑀𝑚=𝑝^′𝑀𝑚+𝑁𝑡=𝑝^′𝑀𝑚+𝑂𝑥;𝑝^′𝑁𝑡=𝑝^′𝑁𝑡+𝑂𝑥	8.1180	0.0078	11	1.6109	0.9515	0.0039	0.0931	0.2973	=𝑝^′𝑀𝑚	=𝑝^′𝑀𝑚	0.5001	0.1189	=𝑝^′𝑁𝑡	0.1588	0.1520
S11	𝑝(𝑁𝑦)(𝐶,𝑖,𝐵,𝑎𝑖,𝑀);𝑝^′𝑂𝑥=𝑝^′𝐿𝑖=0;𝑝^′𝑀𝑚=𝑝^′𝑀𝑚+𝑁𝑡=𝑝^′𝑀𝑚+𝑂𝑥;𝑝^′𝑁𝑡=𝑝^′𝑁𝑡+𝐿𝑖	10.5181	0.0024	11	1.6521	0.9478	0.0042	0.1006	0.3013	=𝑝^′𝑀𝑚	=𝑝^′𝑀𝑚	0.5203	0.1307	0.1590	=𝑝^′𝑁𝑡	0.1711
S12	𝑝(Ny)(𝐶,𝑖,𝐵,𝑎𝑖,𝑀);𝑝^′𝑂𝑥=𝑝^′𝐿𝑖=0;𝑝^′𝑀𝑚=𝑝^′𝑀𝑚+𝑁𝑡=𝑝^′𝑀𝑚+𝑂𝑥;𝑝^′𝑁𝑡=𝑝^′𝑁𝑡+𝑀𝑚	9.2109	0.0045	11	1.5667	0.9461	0.0044	0.0927	0.2874	=𝑝^′𝑀𝑚	=𝑝^′𝑀𝑚	0.4388	0.1238	0.1494	0.1684	=𝑝^′𝑁𝑡
S13	𝐩(𝐍𝐲)(𝐂,𝐢,𝐁,𝐚𝐢,𝐌);𝐩^′𝐎𝐱=𝐩^′𝐋𝐢=𝟎;𝐩^′𝐌𝐦=𝐩^′𝐌𝐦+𝐍𝐭=𝐩^′𝐌𝐦+𝐎𝐱;𝐩^′𝐍𝐭<𝐩^′𝐍𝐭+𝐏𝐣	0.0244	0.4473	10	1.7054	0.9780	0.0018	0.0928	0.2974	=𝑝^′𝑀𝑚	=𝑝^′𝑀𝑚	0.4972	0.1180	0.1750	=𝑝^′𝑁𝑡+𝑂𝑥	=𝑝^′𝑁𝑡+𝑂𝑥
S14	𝐩(𝐍𝐲)(𝐂,𝐢,𝐁,𝐚𝐢,𝐌);𝐩^′𝐎𝐱=𝐩^′𝐋𝐢=𝟎; 𝐩^′𝐌𝐦=𝐩^′𝐌𝐦+𝐏𝐣;𝐩^′𝐍𝐭<𝐩^′𝐍𝐭+𝐏𝐣	0.0000	0.4528	9	1.7292	0.9755	0.0020	0.0929	0.3338	=𝑝^′𝑀𝑚	=𝑝^′𝑀𝑚	=𝑝^′𝑀𝑚	0.1171	0.1722	=𝑝^′𝑁𝑡+𝑂𝑥	=𝑝^′𝑁𝑡+𝑂𝑥
S15	𝑝(Ny)𝐶,𝑖,𝐵,𝑎𝑖;𝑝^′𝑂𝑥=𝑝^′𝐿𝑖=0;𝑝^′𝑀𝑚=𝑝^′𝑀𝑚+𝑃𝑗;𝑝^′𝑁𝑡=𝑝^′𝑁𝑡+𝑃𝑗	7.6905	0.0097	8	1.5302	0.9224	0.0063	0.0981	0.3221	=𝑝^′𝑀𝑚	=𝑝^′𝑀𝑚	=𝑝^′𝑀𝑚	0.1244	=𝑝^′𝑁𝑡	=𝑝^′𝑁𝑡	=𝑝^′𝑁𝑡
S16	𝑝(Ny)(𝐶,𝑖,𝑎𝑖,𝑀);𝑝^′𝑂𝑥=𝑝^′𝐿𝑖=0;𝑝^′𝑀𝑚=𝑝^′𝑀𝑚+𝑃𝑗;𝑝^′𝑁𝑡<𝑝^′𝑁𝑡+𝑃𝑗	25.7016	0.0000	8	1.4843	0.0000	0.0151	0.0899	0.3239	=𝑝^′𝑀𝑚	=𝑝^′𝑀𝑚	=𝑝^′𝑀𝑚	0.0852	0.1631	=𝑝^′𝑁𝑡+𝑂𝑥	=𝑝^′𝑁𝑡+𝑂𝑥
S17	𝑝(Ny)(𝐶)	57.8675	0.0000	3	1.1455	0.0000	0.0813	0.2412	—	—	—	—	—	—	—	—
		𝑐=1.7180

The general model S1, 𝑝_(Ny)(𝐶,𝑖,𝐵_𝑖,𝑎_𝑖,𝑀) all, specifies that: there is a constant unknown mortality of whitefly nymphs in all replicates (𝐶); predation rates on whitefly nymphs varies with predator identity (𝑖), blocks (𝐵_𝑖 indicating that 𝛽_𝑖 varies with predator identity) and whitefly nymph density (𝑎_𝑖 indicating that 𝑎_𝑖 varies with predator identity); species-specific predation rates vary also between single- and multiple-predator treatment (𝑀); parameter structure refers to all four predators, (all). The variation of model structure following each simplification step is: S1 → S2: no predation from O. lineatus is assumed (all → 𝑝^′_𝑂𝑥=0); S2 → S3: no predation from P. phrygianus is further assumed (𝑝^′_𝑂𝑥=0→𝑝^′_𝑂𝑥=𝑝^′_𝐿𝑖=0); S3 → S4: block effect constant between predator species (𝐵_𝑖 → B); S4 → S5: effect of starting whitefly density on predation rates constant between predators (𝑎_𝑖→𝑎); S4 → S6: unknown mortality absent (𝐶 → ⋅); S4 → S7: per capita predation rates of M. melanotoma are identical in Mm and Mm + Nt treatments (𝑝^′_𝑀𝑚=𝑝^′_{𝑀𝑚+𝑁𝑡}), and so on (S7 → S8 → S9; S8 → S10; S8 → S11; S8 → S12; S8 → S13 → S14 → S15; S14 → S16). 𝐾_𝑖: number of model parameters (including the dispersion parameter of the negative binomial distribution, 𝑘_NB, and overdispersion, 𝑐_𝑖). Model selection diagnostics are described in the method section. Parameter estimates are: unknown mortality (𝑝_𝐶Ny) and per capita predation rates of predators in single-(e.g., 𝑝^′_𝑀𝑚 for M. melanotoma) and multiple-predator treatments (e.g., 𝑝^′_{𝑀𝑚+𝑁𝑡} for M. melanotoma when in combination with N. tenuis). The sign — indicates that the parameter has been set to zero. Best models in bold.