Digestive physiology and feeding behaviour of equids a comparative approach

Similar documents
1. Number of Competencies Evaluated. 2. Number of Competencies Rated 2 or Percent of Competencies Attained (2/1) Grade. Instructor Signature

10/3/2016. NRC reqt s for Replacement Ewes. Developing Replacement Ewe Lambs. Differences in Feeding Market Lambs vs Replacement Ewe Lambs

INTRODUCTION TO ANIMAL AND VETERINARY SCIENCE CURRICULUM. Unit 1: Animals in Society/Global Perspective

DOG & CAT CARE & NUTRITION KNOWLEDGE AND RESPECT DOG AND CAT FIRST

A. Body Temperature Control Form and Function in Mammals

Spot the Difference: Using the domestic cat as a model for the nutritional management of captive cheetahs. Katherine M. Bell

Effect of level of intake on methane production per kg of dry matter intake. MAF Technical Paper No: 2011/95

Special Educational Needs (SEN) CARING FOR ANIMALS

Animal Traits and Behaviors that Enhance Survival. Copyright 2010:PEER.tamu.edu

Extra. Feed planning for ewes in late pregnancy and early lactation, during the housed period. Take a stepped approach to feed planning.

RECENT ADVANCES IN OSTRICH NUTRITION IN SOUTH AFRICA: EFFECT OF DIETARY ENERGY AND PROTEIN LEVEL ON THE PERFORMANCE OF GROWING OSTRICHES

The strategic use of injectable trace minerals to improve fertility and health

Module # 1 Component # 7. Mammal Questions. FGASA Exam Prep Course. Copyright. Mammal Questions

Dr. Jerry Shurson Department of Animal Science University of Minnesota

Destination Vet Programme

Animal Sciences (

For Accelerated Rumen Development in Calves

Cattle Egret and Cape Buffalo

2018 ANIMAL HEALTH REGULATIONS FOR FAIRS AND SHOWS IN WISCONSIN

Assessment Schedule 2017 Subject: Agricultural and Horticultural Science: Demonstrate knowledge of livestock management practices (90921)

American Bison (Bison bison)

Bixby Public Schools Course Animal Science Grade: 10,11,12

Snowshoe Hare. Lepus americanus. Other common names. Introduction. Physical Description and Anatomy. Snowshoe rabbit, varying hare, white rabbit

Trevor DeVries Dr. Trevor DeVries is an Associate Professor in the Department of Animal and Poultry Science at the University of Guelph.

GUINEA PIG CARE SHEET

ASSESSMENT. Assessment

2011 The Simple Homeschool Simple Days Unit Studies The Moose

Animal Health Regulations for Fairs and Shows in Wisconsin: 2017 Season

Australian and New Zealand College of Veterinary Scientists. Membership Examination. Veterinary Behaviour Paper 1

Inside This Issue. BEYOND numbers. Small Ruminant

Getting your rabbits. into shape

March 2018 Level 3 Advanced Technical Certificate in Equine Management Level 3 Equine Management Theory Exam (1)

Zurich Open Repository and Archive. To chew or not to chew: fecal particle size in herbivorous reptiles and mammals

Product Catalogue Veterinary Models VETERINARY MODELS VET ACUPUNCTURE MODELS VET ANATOMY MODELS

SKELETONS: Museum of Osteology Tooth and Eye Dentification Teacher Resource

Conservation (last three 3 lecture periods, mostly as a led discussion). We can't cover everything, but that should serve as a rough outline.

Post Mortem Fee Structure

ANIMAL GENETIC RESOURCES IN LATIN AMERICA AND THE CARIBBEAN: UTILIZATION OF BIOTECHNOLOGIES

Year At a Glance HST II Veterinary Assisting Semester Course

Gas emissions according to different pig housing systems

Over feeding rams. Ian McDougall MRCVS. Farmgene Ltd.

Pangolins: 13 facts about the world's most hunted animal by Guy Kelley

FEEDING EWES BETTER FOR INCREASED PRODUCTION AND PROFIT. Dr. Dan Morrical Department of Animal Science Iowa State University, Ames, Iowa

Pre-lab Homework Lab 9: Food Webs in the Wild

Woolly Mammoth & African Elephant

Feeding dairy ewes. Sam Peterson Institute of veterinary, animal and biomedical sciences Massey University

Mental stim ulation it s not just for dogs!! By Danielle Middleton- Beck BSc hons, PGDip CABC

LATE WINTER DIETARY OVERLAP AMONG GREATER RHEAS AND DOMESTIC HERBIVORES ON THE ARGENTINEAN FLOODING PAMPA

Silage Analysis and Ration Planning: Benefits of knowing what you re feeding your stock. Mary McDowell Trainee Livestock Nutritionist

BSc, PhD, MBA, LLM, MSc (Med Sci), DAgVetPharm, FRPharmS, FCPP, FIPharmM, MPS (NZ), FNZCP, FFHom

ADNAN MENDERES UNIVERSITY FACULTY OF VETERINARY MEDICINE

Park Vets Guide to Rabbit Ownership

Make sure you come down to the Zoo of the Wild. On Sundays everybody is HALF OFF!

1. Hair 2. Mammary glands produce milk 3. Specialized teeth 4. 3 inner ear bones 5. Endothermic 6. Diaphragm 7. Sweat, oil and scent glands 8.

Sheep Care on Small Farms and Homesteads

Appendix I Average Analyses of B.C. Feeds

Phase B 5 Questions Correct answers are worth 10 points each.

[Rev. 2012] CAP. 364 Animal Diseases

Main Points. 2) The Great American Interchange -- dispersal versus vicariance -- example: recent range expansion of nine-banded armadillos

Alachua County Youth Fair Cattleman s Study Guide

Before James Hunt built the Covered Bridge and named the city Coral Springs in

Table1. Target lamb pre-weaning daily live weight gain from grazed pasture

Planning Spring/Summer 2018

TACKLING THE PARASITOLOGICAL CHALLENGES ARISING FROM ORGANIC FARMING PRACTICES

Pedigree ADVANCE Add life to the life of your dog Total care for your puppy vitality and health Choosing, your puppy

Your dog a guide to feeding dogs aged 1-6

Mona I. Mohammady, A.H. Hammam and N. H. Ibrahim

Main Points. 2) The Great American Interchange -- dispersal versus vicariance -- example: recent range expansion of nine-banded armadillos

Main Points. 2) The Great American Interchange -- dispersal versus vicariance -- example: recent range expansion of nine-banded armadillos

Iguana Husbandry, Nutrition and Disease

Animal Care & Ethics Committee

Animals WORKSHEET 3.1 Animals

Small Animal Vet Science Chilton High School

Fish 475: Marine Mammalogy

VETERINARY MEDICINE-VM (VM)

BETTERRETURNS. Feeding the ewe A manual for consultants, vets and producers

Appendix: Hollow Dinosaurs?

Animal Welfare Assessment and Challenges Applicable to Pregnant Sow Housing

TCHS ANIMAL SCIENCE. 100 LEVEL COMPETENCY TASKS Small Animal Care & Management Auth: Dean M. Warren CIP CODE: ANIMAL SCIENCES, GENERAL

The complete guide to. Puppy Growth Charts. Puppy Growth Chart. Puppy Growth Chart. Dog s Name: Dog s Name: D.O.B. Dog s Name: Neuter Date:

ISAAT species list. Preamble

SHEEP BRP MANUAL 12. Improving ewe nutrition for Better Returns

Pricelist Veterinary Models VETERINARY MODELS VET ACUPUNCTURE MODELS VET ANATOMY MODELS

EQUINE CUSHING S DISEASE

Internal Assessment Resource NCEA Level 1 Science AS KEEP CALM AND COUNT SHEEP. A unit of learning to be assessed for

Global Perspective of Rabies. Alexander I. Wandeler CFIA Scientist Emeritus

Hy D. for Layers and Breeders. DSM Nutritional Products

Stark County Rabies Prevention Information Manual

PORCINE ZONA PELLUCIDA IMMUNOCONTRACEPTION OF SOUTHERN WHITE RHINOCEROS IN MANAGED CARE.

Key Information. Mountain Hill Vs Lowland Production. Breeding Strategy

STUDENT QUESTIONS & ANSWERS: GRADE 1 & 2

Animal Study: Adelaide Zoo

Question Set 1: Animal EVOLUTIONARY BIODIVERSITY

ESRM 350 The Decline (and Fall?) of the White-tailed Jackrabbit

Subject: Animal Science Calendar : Timeframe: 1 st 9 Weeks

Managing pre-calving dairy cows: nutrition, housing and parasites

Beef... Beef Natural Contamination... Beef liver...

SHORT DESCRIPTION OF TECHNICAL PAPER CONTENT

Jefferson County High School Course Syllabus

PRESENTATION OF FINDINGS ARTIFICIAL INSEMINATION BUSINESS MODEL ASSESSMENT

Transcription:

Digestive physiology and feeding behaviour of equids a comparative approach Marcus Clauss Clinic for Zoo Animals, Exotic Pets and Wildlife, Vetsuisse Faculty, University of Zurich, Switzerland Gent 2013

8 species Large herbivore diversity: Equids

Digestive tract: Equids from Stevens und Hume (1995), Clauss et al. (2008)

Large herbivore diversity: Perissodactyls 16 species

Digestive tract: Perissodactyls from Stevens und Hume (1995), Clauss et al. (2008), Müller et al. (in prep.)

Large herbivore diversity: hoofed mammals 16 species > 300 species

Digestive tract: Perissodactyls vs. Artiodactyls Stevens und Hume (1995), Clauss et al. (2008), Schwarm et al. (2010), Müller et al. (in prep.)

Large herbivore diversity: hoofed mammals 16 species > 300 species

Large herbivore diversity: Europe

Large herbivore diversity: Europe

Large herbivore diversity in deep time: Perissodactyls

Large herbivore diversity in deep time: Perissodactyls Chalicotheres

Large herbivore diversity in deep time: Perissodactyls Brontotheres Chalicotheres

Large herbivore diversity in deep time: Perissodactyls Brontotheres Chalicotheres

from Janis et al. (1994)

from Janis et al. (1994)

from Janis et al. (1994)

from Janis et al. (1994)

from Janis et al. (1994)

from Janis et al. (1994)

from Janis et al. (1994)

from Janis et al. (1994)

from Janis et al. (1994)

from Janis et al. (1994)

from Janis et al. (1994)

from Janis et al. (1994)

from Janis et al. (1994)

from Janis et al. (1994)

from Janis et al. (1994)

Large herbivore diversity: hoofed mammals 16 species > 300 species

Large herbivore diversity: hoofed mammals 16 species > 300 species

Mammal gestation period For any mammal, achieving the same degree of neonatal development in a shorter gestation period if not associated with higher costs should be advantageous (higher fecundity due to shorter generation times).

Mammal gestation period For any mammal, achieving the same degree of neonatal development in a shorter gestation period if not associated with higher costs should be advantageous (higher fecundity due to shorter generation times). Days of gestation period (to apparently similar level of precociality) Cattle: app. 280 days Horse: app. 340 days Dromedary: app. 390 days Okapi: app. 440 days

Mammal gestation period For any mammal, achieving the same degree of neonatal development in a shorter gestation period if not associated with higher costs should be advantageous (higher fecundity due to shorter generation times). Days of gestation period (to apparently similar level of precociality) Cattle: app. 280 days Horse: app. 340 days Dromedary: app. 390 days Okapi: app. 440 days The difference cannot be due to body size!

Mammal gestation period For any mammal, achieving the same degree of neonatal development in a shorter gestation period if not associated with higher costs should be advantageous (higher fecundity due to shorter generation times). Days of gestation period (to apparently similar level of precociality) Cattle: app. 280 days Horse: app. 340 days Dromedary: app. 390 days nearly extinct in a Okapi: app. 440 days very limited geographical range

Mammal gestation period For any mammal, achieving the same degree of neonatal development in a shorter gestation period if not associated with higher costs should be advantageous (higher fecundity due to shorter generation times). Days of gestation period (to apparently similar level of precociality) Cattle: app. 280 days Horse: app. 340 days only in extreme, Dromedary: app. 390 days resource-poor Okapi: app. 440 days habitats

Mammal gestation period For any mammal, achieving the same degree of neonatal development in a shorter gestation period if not associated with higher costs should be advantageous (higher fecundity due to shorter generation times). Days of gestation period (to apparently similar level of precociality) Cattle: app. 280 days rule the world!! Horse: app. 340 days Dromedary: app. 390 days Okapi: app. 440 days

Mammal gestation period For any mammal, achieving the same degree of neonatal development in a shorter gestation period if not associated with higher costs should be advantageous (higher fecundity due to shorter generation times). Days of gestation period (to apparently similar level of precociality) Cattle: app. 280 days Horse: app. 340 days Dromedary: app. 390 days Okapi: app. 440 days We would predict that animals with a shorter gestation period should be particularly successful (e.g. in terms of species diversity).

(Precocial) Mammal gestation period 1000 Gestation period (d) 100 10 1 100 10000 1000000 100000000 Body mass (g) from Clauss et al. (2013)

(Precocial) Mammal gestation period 1000 Gestation period (d) 100 10 1 100 10000 1000000 100000000 Body mass (g) from Clauss et al. (2013)

(Precocial) Mammal gestation period 1000 Gestation period (d) 100 10 1 100 10000 1000000 100000000 Body mass (g) from Clauss et al. (2013)

Mammal chewing efficiency For any herbivore, increasing chewing efficiency if not associated with higher costs should be advantageous (higher feeding efficiency due to higher digestibility) because there is

Mammal chewing efficiency For any herbivore, increasing chewing efficiency if not associated with higher costs should be advantageous (higher feeding efficiency due to higher digestibility) because there is

Mammal chewing efficiency For any herbivore, increasing chewing efficiency if not associated with higher costs should be advantageous (higher feeding efficiency due to higher digestibility) because there is and therefore

Mammal chewing efficiency For any herbivore, increasing chewing efficiency if not associated with higher costs should be advantageous (higher feeding efficiency due to higher digestibility) because there is and therefore

Mammal chewing efficiency 100 MPS (mm) 10 1 0.1 0.001 0.01 0.1 1 10 100 1000 10000 BM (kg) Simple-stomached Nonruminant ff from Fritz et al. (2009)

Mammal chewing efficiency 100 MPS (mm) 10 1 0.1 0.001 0.01 0.1 1 10 100 1000 10000 BM (kg) Simple-stomached Nonruminant ff from Fritz et al. (2009)

Mammal chewing efficiency 100 MPS (mm) 10 1 0.1 0.001 0.01 0.1 1 10 100 1000 10000 BM (kg) Simple-stomached Nonruminant ff Ruminants from Fritz et al. (2009)

Large mammal molar surfaces from Jernvall et al. (1996)

Large mammal molar surfaces from Jernvall et al. (1996)

Large mammal molar surfaces from Jernvall et al. (1996)

Large mammal molar surfaces from Jernvall et al. (1996), Schwarm et al. (2008)

Ruminant sorting mechanism from Jernvall et al. (1996), Schwarm et al. (2008)

Parallel evolution? Perissodactyls Artiodactyls

Parallel evolution? among Perissodactyls Artiodactyls

Parallel evolution? among Perissodactyls Artiodactyls Equids Ruminants

Parallel evolution? among Perissodactyls Artiodactyls Equids Ruminants achieve comparatively high chewing efficiencies and food intakes

Parallel evolution? among Perissodactyls Artiodactyls Equids Ruminants achieve comparatively high chewing efficiencies and food intakes but

Advantage ruminants? Due to their superior chewing efficiency, ruminants achieve higher digestibilities... from Clauss et al. (2009; data from Foose 1982)

Advantage ruminants? Due to their superior chewing efficiency, ruminants achieve higher digestibilities... from Clauss et al. (2009; data from Foose 1982)

Advantage ruminants? Due to their superior chewing efficiency, ruminants achieve higher digestibilities... and therefore do not require as high a food intake.

Advantage ruminants? Due to their superior chewing efficiency, ruminants achieve higher digestibilities... and therefore do not require as high a food intake.

Advantage ruminants? Because they need to feed more (and do not have rumination breaks ), equids nearly feed continuously.

Advantage ruminants? Because they need to feed more (and do not have rumination breaks ), equids nearly feed continuously. Grazing

Advantage ruminants? Because they need to feed more (and do not have rumination breaks ), equids nearly feed continuously. Grazing

Advantage ruminants? Because do not have rumination breaks, equids have higher ingestive mastication activity.

Advantage ruminants? Are horses more susceptible to low food intake than ruminants?

Advantage ruminants? Horses cannot achieve the difference between particle and fluid retention (SF selectivity factor) as observed in ruminants.

Advantage ruminants? In spite of theoretical concept

Advantage ruminants? In spite of theoretical concept no net empirical indication for differentiated passage in horses. from Lechner et al. (2010), Clauss et al. (pers. obs.)

Other differences: Calcium digestibility

Other differences: Calcium digestibility Phosphorus is supplied directly to microbes via saliva P from Stevens & Hume (1995)

Other differences: Calcium digestibility Phosphorus is supplied directly to microbes via saliva P Ca In order to guarantee phosphorus availability in the hindgut, calcium is actively absorbed from ingesta and excreted via urine from Stevens & Hume (1995) hypothesis by Clauss & Hummel (2008)

Why equids? Other perissodactyls survive in body size ranges beyond the ruminant range (rhinos) or in absence of ruminant competition (tapirs). Why / how do equids survive (only in the upper ruminant body size range, and only in the grazing niche)?

Other differences: Methane production? Methane (l/d) 1000 100 10 1 Ruminants Camelid Equids Elephant Wallaby Hyrax Rabbit Guinea pig 0.1 0.01 0.1 1 10 100 1000 10000 Body mass (kg) from Franz et al. (2010)

Other differences: Methane production? Methane (l/d) 1000 100 10 1 Ruminants Camelid Equids Elephant Wallaby Hyrax Rabbit Guinea pig 0.1 0.01 0.1 1 10 100 1000 10000 Body mass (kg) from Franz et al. (2010)

Other differences: Methane production? Methane (l/d) 1000 100 10 1 Ruminants Camelid Equids Elephant Wallaby Hyrax Rabbit Guinea pig 0.1 0.01 0.1 1 10 100 1000 10000 Body mass (kg) from Franz et al. (2010)

Other differences: Methane production? Methane (%DE) 100 10 1 Ruminants Equids Rabbits Guinea pigs 0.1 0.1 1 10 100 1000 Body mass (kg) from Franz et al. (2010)

Equid-ruminant facilitation?

Digestive advantage for equids? When resources are scarce on African game farms, the ruminants lose condition first but it is when the zebras lose condition that you need to start to worry. (Adrian Shrader, pers. comm.)

The traditional view of foregut vs. hindgut fermentation Janis (1976)

Do you believe it? if diet quality gets lower, a horse simply eats more

Data in sheep

The traditional view of foregut vs. hindgut fermentation

160 140 120 DMI (g kg -0.75 d -1 ) 100 80 60 40 20 Sheep Sheep (this study) Mountain sheep 0 10 20 30 40 50 60 70 80 90 Forage NDF (%DM)

160 140 120 DMI (g kg -0.75 d -1 ) 100 80 60 40 20 Sheep Sheep (this study) Mountain sheep 0 10 20 30 40 50 60 70 80 90 Forage NDF (%DM)

160 140 120 DMI (g kg -0.75 d -1 ) 100 80 60 40 20 Sheep Sheep (this study) Mountain sheep 0 10 20 30 40 50 60 70 80 90 Forage NDF (%DM)

DMI (g kg -0.75 d -1 ) 160 140 120 100 80 60 40 Sheep Cattle Bison Banteng/Gaur Yak Water buffalo S. caffer Muskox 20 0 10 20 30 40 50 60 70 80 90 Forage NDF (%DM)

DMI (g kg -0.75 d -1 ) 160 140 120 100 80 60 40 Sheep Cattle Bison Banteng/Gaur Yak Water buffalo S. caffer Muskox 20 0 10 20 30 40 50 60 70 80 90 Forage NDF (%DM)

DMI (g kg -0.75 d -1 ) 160 140 120 100 80 60 40 Sheep Cattle Bison Banteng/Gaur Yak Water buffalo S. caffer Muskox 20 0 10 20 30 40 50 60 70 80 90 Forage NDF (%DM)

DMI (g kg -0.75 d -1 ) 160 140 120 100 80 60 40 Sheep Cattle Donkey 20 0 10 30 50 70 90 Forage NDF (%DM) Meyer et al. (2010)

DMI (g kg -0.75 d -1 ) 160 140 120 100 80 60 40 Sheep Cattle Donkey 20 0 10 30 50 70 90 Forage NDF (%DM) Meyer et al. (2010)

DMI (g kg -0.75 d -1 ) 160 140 120 100 80 60 40 Sheep Cattle Horses 20 0 10 30 50 70 90 Forage NDF (%DM) Meyer et al. (2010)

DMI (g kg -0.75 d -1 ) 160 140 120 100 80 60 40 Sheep Cattle Horses 20 0 10 30 50 70 90 Forage NDF (%DM) Meyer et al. (2010)

DMI (g kg -0.75 d -1 ) 160 140 120 100 80 60 40 Sheep Cattle Horses Wild equids 20 0 10 30 50 70 90 Forage NDF (%DM) Meyer et al. (2010)

DMI (g kg -0.75 d -1 ) 160 140 120 100 80 60 40 Sheep Cattle Horses Wild equids DMI (g kg -0.75 d -1 ) 160 140 120 100 80 60 40 Sheep Cattle White rhino Indian rhino Black rhino 20 20 0 10 30 50 70 90 Forage NDF (%DM) 0 10 30 50 70 90 Forage NDF (%DM)

DMI (g kg -0.75 d -1 ) 160 140 120 100 80 60 40 Sheep Cattle Horses Wild equids DMI (g kg -0.75 d -1 ) 160 140 120 100 80 60 40 Sheep Cattle White rhino Indian rhino Black rhino 20 20 0 10 30 50 70 90 Forage NDF (%DM) 0 10 30 50 70 90 Forage NDF (%DM)

Misinterpretation of anatomical features? DMI (g kg -0.75 d -1 ) 160 140 120 100 80 60 40 Sheep Cattle Horses Wild equids DMI (g kg -0.75 d -1 ) 160 140 120 100 80 60 40 Sheep Cattle White rhino Indian rhino Black rhino 20 20 0 10 30 50 70 90 Forage NDF (%DM) 0 10 30 50 70 90 Forage NDF (%DM) from Stevens und Hume (1995), Clauss et al. (2008)

The traditional view of foregut vs. hindgut fermentation Janis (1976)

Extant horses are grazers

Diet and mesowear Kaiser and Fortelius (2003)

Diet and mesowear

Diet and mesowear

Adapted to abrasive diets

Diet and mesowear: zoo vs. wild 2.00 1.50 Difference in score 1.00 0.50 0.00-0.50 Less abrasion than in the wild More abrasion than in the wild -1.00-1.50 from Kaiser et al. (2009)

Diet and mesowear: zoo vs. wild 2.00 1.50 Difference in score 1.00 0.50 0.00-0.50 Less abrasion than in the wild More abrasion than in the wild -1.00-1.50 from Kaiser et al. (2009), Taylor et al. (in prep.)

Wild equids in captivity Similar as in grazing ruminants, few health problems related to nutrition in captive wild equids (because zoo diets are typically more forage dominated?)

Wild equids in captivity Similar as in grazing ruminants, few health problems related to nutrition in captive wild equids (because zoo diets are typically more forage dominated?) - Incidents of dental abnormalities (a) Free-ranging (b) Captive from Taylor et al. (in prep.)

Wild equids in captivity Similar as in grazing ruminants, few health problems related to nutrition in captive wild equids (because zoo diets are typically more forage dominated?) - Incidents of dental abnormalities Hoyer et al. (2012)

Wild equids in captivity Similar as in grazing ruminants, few health problems related to nutrition in captive wild equids (because zoo diets are typically more forage dominated?) - Incidents of dental abnormalities - Hoof overgrowth/laminitis

Equid seasonality

Wild equids in captivity Similar as in grazing ruminants, few health problems related to nutrition in captive wild equids (because zoo diets are typically more forage dominated?) - Incidents of dental abnormalities - Hoof overgrowth/laminitis - Obesity

Equid seasonality

Equid seasonality

Equid seasonality

Wild equids in captivity Similar as in grazing ruminants, few health problems related to nutrition in captive wild equids (because zoo diets are typically more forage dominated?) - Incidents of dental abnormalities - Hoof overgrowth/laminitis - Obesity - Colic - Vitamin E deficiency

Conclusion From a nutritional point of view, wild equids appear well understood: (grass) forage diets available at all times, ideally with mimicking seasonal patterns in the wild. How the digestive physiology of equids differs from that of ruminants, especially in terms of minimum intake tolerable and differential digesta movements, remain to be investigated. The sequence, and the mechanisms, of the equid-ruminant diversification and competition in evolution remain to be explained in a way that matches empirical data.

thank you for your attention