Overview of WinEquus

Stephen JenkinsEmeritus Professor of Biology

University of Nevada, Renojenkins@unr.edu

Two Purposes of Modeling(Caswell 1976)

• Models for understanding

• Models for prediction ≈ forecasting

• Doomsday: 2026.87 = 13 November 2026(von Foerster et al. 1960, Science 132:1291)

Caswell, H. 1976. The validation problem. Pages 313-325 in B. C. Patten, editor. Systems analysis and simulation in ecology. Academic Press, New York, New York, USA.

Data Requirements for WinEquus

• initial age-sex distribution• annual survival probabilities for each age-sex class• annual foaling rates for each age class of mares• sex ratio at birth

• ideally, these data should– be site-specific– have variance estimates

Data Available in Practice

• estimate of population size at a site• sometimes, estimate of age-sex distribution

– if horses released in recent gather were aged

• sex ratio near birth?

• estimates of survival and reproduction at a few sites– 11 years of data for Pryor Mountain, MT (Garrott & Taylor 1990)

– 6 years for the Granite Range, NV (Berger 1986)

– 7 years for Garfield Flat, NV (Ashley 2000)

Is lack of site-specific data a problem?

Is lack of site-specific data a problem?

Is lack of site-specific data a problem?

• recent data → average annual adult survival > 90% for

– Cumberland Island, GA (Goodloe et al., 2000, J. Wildl. Manage. 64:114-121)

– Montgomery Pass, NV-CA(Turner & Morrison, 2001, Southwestern Naturalist 46:183-190)

– Kaimanowa Ranges, New Zealand(Linklater et al., 2004, Wildl. Res. 31:119-128)

– Przewalski’s wild horses in France (not free-ranging)(Tatin et al., 2008, J. Zool. 277:134-140)

– recently feral horses in the Camargue in France(Grange et al., 2009, Proc. Royal Soc. B 276:1911-1919)

– Tornquist Park, Argentina(Scorolli & Lopez Cazorla, 2010, Wildl. Res. 37:207-214)

Is lack of site-specific data a problem?

• There is more variation between sites in

– annual survival probability of foals

– foaling rate, especially of young mares

Location Foaling Rate of 2-year-olds

Pryor Mountain, 1976-1986 0

Pryor Mountain, 1996-2000 0.08

Granite Range, 1979-1983 0.35

Garfield Flat, 1993-1999 0.52

Three kinds of stochasticity in WinEquus

• Measurement uncertainty in initial population size

• Demographic stochasticity

• Environmental stochasticity

Measurement uncertainty in initial population size: User adjustments

– 90% sighting probability is default(Garrott et al. 1991. J. Wildl. Manage. 55:641-648)

– WinEquus uses abeta-binomial model

– User may specifyexact initial conditionsinstead

Demographic Stochasticity in WinEquus:User adjustments

None …

… foaling rate = 0.5→ 50% chance of foaling for each mare

→ 10 mares may have 5 foals,or 4 or 6, or 3 or 7, or 2 or 8, …

Environmental Stochasticity in WinEquus:User adjustments

• Eleven years of data for Pryor Mountain, MT(Garrott and Taylor. 1990. J. Wildl. Manage. 54:603-612)

→ Logistic distributions used to simulate stochasticity

Garfield Flat, NV:

selective removal in winter 1997 → non-equilibrium age distribution

N0 = 109

Density-Dependence in WinEquus

Density-Dependence in WinEquus

Why no density-dependence as default?

(1)Populations often managed below levels where DD effects likely.

(2)Insufficient data to estimate carrying capacity or form of DD effects.

Some thoughts on density-dependence

1) Experimental evidence

2) Analysis of short time series

3) Predation may regulate feral horse populations in some places

4) Without predators, carrying capacity for some herbivores may mean high mortality or habitat degradation

Experimental evidence of density-dependence:feral donkeys in Australia (Choquenot, 1991, Ecology 72:805-813)

Culled Population Control Population

Density reduction 80% --

Density after 4 yrs. 1.8/km2 3.2/km2

Growth rate (r) 0.18 0

Foaling rate 87% 77%

Foal mortality rate 21% 62%

Short time series & density dependence(Grange et al., 2009, Proc. Royal Soc. B. 276:1911-1919, Scorolli & Lopez Cazorla, 2010,

Wildl. Res. 37:207-214)

Predation & density dependence (Turner & Morrison, 2001, SW Naturalist 46:183-190)

Mountain lions may regulate horse populations e.g., at Montgomery Pass, mountain lions killed 45% of foals/yr

this 22 mi2 reserve is productive (high ppt, fertile soil)

a rewilding experiment (without mammalian predators)1983-1992: 32 Heck cattle, 20 konik ponies, 44 red deerintroduced2010: 3000 ungulates; 15-24% of horses starve in winter

What happens at carrying capacity?

Oostvaardersplassen, the Netherlands

(F. W. M. Vera, June 2009, British Wildlife)

Two Purposes of Modeling(Caswell 1976)

• Models for understanding

• Models for prediction ≈ forecasting

Caswell, H. 1976. The validation problem. Pages 313-325 in B. C. Patten, editor. Systems analysis and simulation in ecology. Academic Press, New York, New York, USA.

A WinEquus Example

• How would fertility control affect population growth at Garfield Flat?

• Initial conditions and assumptions– N0 ≈ 109, like after selective removal of young horses in Feb

1997– average survival probabilities, foaling rates, sex ratio @

birth as found by Ashley & Jenkins for 1993-1999– year-to-year variation in survival and foaling follow logistic

distributions

Ave

rag

e A

nn

ua

l Gro

wth

Ra

te (

%)

Cumulative Percentage of Trials

0

5

10

15

0 20 40 60 80 100

Ave

rag

e A

nn

ua

l Gro

wth

Ra

te (

%)

Cumulative Percentage of Trials

0

5

10

15

20

25

0 20 40 60 80 100

Growth rates with fertility treatment

Minimum 3.8%

Median 9.7%

Maximum 12.4%

Growth rates without fertility treatment

Minimum 14%

Median 18.3%

Maximum 22.3%