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

Table 2

Best model structure and main parameter estimates relative to enemy biodiversity effects on whitefly nymph survival, adult emergence and survival (joint analysis).

Code	Model structure	Model selection diagnostics					Maximum likelihood estimates of model parameters
		ΔQAICc	𝑤𝑖	𝐾𝑖	𝑐𝑖	𝑅2𝑖	𝑘NB	ΨA	ΨB	𝑝^′𝑂𝑥	𝑝^′𝑂𝑥+𝑁𝑡	𝑝^′𝑂𝑥+𝐿𝑖	𝑝^′𝑂𝑥+𝑀𝑚	𝑝^′𝐿𝑖	𝑝^′𝐿𝑖+𝑁𝑡	𝑝^′𝐿𝑖+𝑂𝑥	𝑝^′𝐿𝑖+𝑀𝑚
J1	𝑝(Ny)(S13) Ψ(𝐵) 𝑝(Ad)(𝐶,𝑖,𝐵𝑖,𝑎𝑖,𝑀)  all	64.6797	0.000	39	1.5351	0.1166	0.2077	0.4466	0.2272	0.2720	0.0000	0.1750	0.0000	0.5198	0.0000	0.0000	0.0000
J2	𝑝(Ny)(S13) Ψ(𝐵) 𝑝(Ad)(𝑖,𝐵𝑖,𝑎𝑖,𝑀)  all	61.5743	0.000	38	1.5255	0.1166	0.2077	0.4466	0.2272	0.2720	0.0000	0.1750	0.0000	0.5198	0.0000	0.0000	0.0000
J3	𝑝(Ny)(S13) Ψ(𝐵) 𝑝(Ad)(𝑖,𝑀)  all	38.1612	0.000	30	1.4528	0.1166	0.2077	0.4473	0.2269	0.2725	0.0000	0.1751	0.0000	0.2745	0.0152	0.0000	0.0000
J4	𝑝(Ny)(S13) Ψ(𝐵) 𝑝(Ad)(𝑖,𝑀);𝑝^′𝑁𝑡=0	27.2339	0.000	26	1.4190	0.1166	0.2077	0.4473	0.2269	0.2725	0.0000	0.0962	0.0000	0.2745	0.0815	0.0873	0.0000
J5	𝑝(Ny)(S13) Ψ(𝐵) 𝑝(Ad)(𝑖,𝑀);𝑝^′𝑁𝑡=𝑝^′𝑀𝑚=0	16.8063	0.000	22	1.3867	0.1166	0.2077	0.4473	0.2269	0.2725	0.0000	0.0918	0.0000	0.2745	0.0815	0.0918	0.0000
J6	𝑝(Ny)(S13) Ψ(𝐵) 𝑝(Ad)(𝑖);𝑝^′𝑁𝑡=𝑝^′𝑀𝑚=0;𝑝^′𝑂𝑥=𝑝^′𝑂𝑥+𝑃𝑗;𝑝^′𝐿𝑖=𝑝^′𝐿𝑖+𝑃𝑗	5.6330	0.016	16	1.3440	0.0109	0.2326	0.4310	0.2234	0.0000	=𝑝^′𝑂𝑥	=𝑝^′𝑂𝑥	=𝑝^′𝑂𝑥	0.0498	=𝑝^′𝐿𝑖	=𝑝^′𝐿𝑖	=𝑝^′𝐿𝑖
J7	𝑝(Ny)(S13) Ψ(𝐵) 𝑝(Ad)(𝑖);𝑝^′𝑁𝑡=𝑝^′𝑀𝑚=0;𝑝^′𝑂𝑥=0;𝑝^′𝐿𝑖=𝑝^′𝐿𝑖+𝑃𝑗	3.2648	0.051	15	1.3367	0.0109	0.2326	0.4310	0.2234	—	—	—	—	0.0498	=𝑝^′𝐿𝑖	=𝑝^′𝐿𝑖	=𝑝^′𝐿𝑖
J8	𝐩(𝐍𝐲) (S13) Ψ(𝐁) 𝐩(𝐀𝐝)(𝐌);𝐩^′𝐍𝐭=𝐩^′𝐌𝐦=𝟎;𝐩^′𝐎𝐱=𝐩^′𝐋𝐢	0.3907	0.216	15	1.3350	0.0886	0.2143	0.4411	0.2249	0.2642	—	—	—	=𝑝^′𝑂𝑥	—	—	—
J9	𝐩(𝐍𝐲) (S13) Ψ(𝐁) 𝐩(𝐀𝐝)(𝐌);𝐩^′𝐍𝐭=𝐩^′𝐌𝐦=𝟎;𝐩^′𝐎𝐱=𝐩^′𝐋𝐢=𝐩^′𝐋𝐢+𝐎𝐱	0.0000	0.263	15	1.3351	0.1037	0.2108	0.4443	0.2266	0.2395	—	—	—	=𝑝^′𝑂𝑥	—	=𝑝^′𝑂𝑥	—
J10	𝐩(𝐍𝐲) (S13) Ψ(𝐁) 𝐩(𝐀𝐝)(𝐌);𝐩^′𝐍𝐭=𝐩^′𝐌𝐦=𝟎; 𝐩^′𝐎𝐱=𝐩^′𝐋𝐢=𝐩^′𝐎𝐱+𝐋𝐢=𝐩^′𝐋𝐢+𝐎𝐱	1.0172	0.158	15	1.3362	0.0734	0.2179	0.4394	0.2258	0.1484	—	=𝑝^′𝑂𝑥	—	=𝑝^′𝑂𝑥	—	=𝑝^′𝑂𝑥	—
J11	𝐩(𝐍𝐲) (S13) Ψ(𝐁) 𝐩(𝐀𝐝)(𝐤)	1.0771	0.153	14	1.3297	0.0000	0.2352	0.4275	0.2223	—	—	—	—	—	—	—	—
J12	𝑝(Ny)(S14) Ψ(𝐵) 𝑝(Ad)(𝑀);𝑝^′𝑁𝑡=𝑝^′𝑀𝑚=0;𝑝^′𝑂𝑥=𝑝^′𝐿𝑖=𝑝^′𝐿𝑖+𝑂𝑥	1.2322	0.142	14	1.3455	0.1060	0.2102	0.4441	0.2264	0.2389	—	—	—	=𝑝^′𝑂𝑥	—	=𝑝^′𝑂𝑥	—
J13	𝑝(Ny)(S13) Ψ(𝐵) 𝑝(Ad)(𝐶)	3.4194	0.045	15	1.3370	0.0000	0.2352	0.4275	0.2223	—	—	—	—	—	—	—	—
J14	𝑝(Ny)(S13) Ψ(⋅) 𝑝(Ad)(𝑀);𝑝^′𝑁𝑡=𝑝^′𝑀𝑚=0;𝑝^′𝑂𝑥=𝑝^′𝐿𝑖=𝑝^′𝐿𝑖+𝑂𝑥	45.7700	0.000	14	1.3063		0.3911	0.3344	=Ψ𝐴	0.2036	—	—	—	=𝑝^′𝑂𝑥	—	=𝑝^′𝑂𝑥	—
		𝑐=1.3362
				ML estimates of parameters relative to predation on nymphs for Model J9
				𝛽	𝑝𝐶Ny		𝑘NB	𝑎𝑁𝑡	𝑎𝑀𝑚	𝑝^′𝑀𝑚	𝑝^′𝑀𝑚+𝑁𝑡	𝑝^′𝑀𝑚+𝑂𝑥	𝑝^′𝑀𝑚+𝐿𝑖	𝑝^′𝑁𝑡	𝑝^′𝑁𝑡+𝑂𝑥	𝑝^′𝑁𝑡+𝐿𝑖	𝑝^′𝑁𝑡+𝑀𝑚
				−0.0892	0.0935		0.0019	0.0015	0.0076	0.2563	=𝑝^′𝑀𝑚	=𝑝^′𝑀𝑚	0.4286	0.1137	0.1686	=𝑝^′𝑁𝑡+𝑂𝑥	=𝑝^′𝑁𝑡+𝑂𝑥

The general model J1, 𝑝_(Ny)(S13) Ψ(𝐵) 𝑝_(Ad)(𝐶,𝑖,𝐵_𝑖,𝑎_𝑖,𝑀)  all, specifies that: model structure for whitefly nymph survival is as in S13, 𝑝_(Ny)(S13); emergence rates vary with blocks, Ψ(𝐵); relatively to whitefly adult survival, 𝑝_(Ad)(⋯), there is a constant unknown mortality in all replicates (𝐶); predation varies with predator identity (𝑖), blocks (𝐵_𝑖 indicating that 𝛽_𝑖 varies with predator identity) and whitefly 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: J1 → J2: unknown mortality absent (𝐶 → ⋅); J2 → J3: block and starting whitefly density effects absent (𝐵_𝑖,𝑎_𝑖 → ⋅); J3 → J4: no predation from N. tenuis is assumed (all → 𝑝^′_𝑁𝑡=0); J4 → J5: no predation from M. melanotoma is further assumed (𝑝^′_𝑁𝑡=0→𝑝^′_𝑁𝑡=𝑝^′_𝑀𝑚=0), and so on (J5 → J6; J5 → J7; J5 → J8; J5 → J9; J5 → J10; J9 → J12; J9 → J13). 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 presented are: emergence rates for block A and B (Ψ_A and Ψ_B) and per capita predation rates of spiders in single- (e.g., 𝑝^′_𝑂𝑥 for O. lineatus) and multiple-predator treatments (e.g., 𝑝^′_{𝑂𝑥+𝐿𝑖} for O. lineatus when in combination with P. phrygianus). At the bottom of the table is also presented the full set of parameter estimates from model J9 relative to predation on whitefly nymphs. The sign — indicates that the parameter has been set to zero. Best models in bold.