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Table 1 The most parsimonious SETAR models of the population dynamics of wolves on Isle Royale, Michigan, 1959–99. Covariates included winter snow accumulation (SW t ), Northern Hemisphere temperature anomalies (T t ), number of packs in the previous year (PAt-1), and mean pack size in the previous year (PSt-1). X t is loge-transformed density, and a i,j are statistical parameters (i = 1 and 2 corresponds to lower and upper regimes, respectively, j = 0, 1, 2, 3 correspond to the constant, lag-1 density coefficients, number of packs coefficients, and climatic coefficients, respectively). n indicates the number of data points in each regime. Parameter estimates were obtained by the method of conditional least squares. X* is the equilibrium point on the log-scale.

From: Population response to climate change: linear vs. non-linear modeling approaches

 

Full Model

θ

Coefficients

SE

p

n

AIC

R 2

Equilibrium

 

X t = a1,0 + a1,1Xt-1+ a1,2PAt-1+ a1,3T t + a1,4SW t

Xt-1< 3.40

a1,0 = 1.89

0.43

0.0002

32

14.18

0.58

X* = 2.40

Non-Linear Model

  

a1,1 = 0.50

0.14

0.001

   

Stable

   

a1,2 = -0.06

0.04

0.17

    
   

a1,3 = -0.47

0.16

0.009

    
   

a1,4 = -.0008

0.0005

0.15

    
 

X t = a2,0 + a2,1Xt-1+ a2,2PSt-1+ a2,3PAt-1+ a2,4SW t

Xt-1≥ 3.40

a2,0 = -5.82

1.43

0.03

8

 

0.96

X* = 3.58

   

a2,1 = 2.63

0.46

0.01

   

Unstable

   

a2,2 = 0.06

0.03

0.11

    
   

a2,3 = -0.24

0.05

0.02

    
   

a2,4 = 0.002

0.0009

0.12

    

Linear Model

X t = a0+ a1Xt-1+ a2PAt-1+ a3T t

 

a0 = 0.86

0.33

0.01

40

27.98

0.69

X* = 2.97

   

a1 = 0.81

0.12

0.00

   

Stable

   

a2 = -0.07

0.03

0.06

    
   

a3 = -0.39

0.18

0.04

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