Tree Harvesting In Age-structured Forests Subject To Beetle Infestations

Size: px

Start display at page:

Download "Tree Harvesting In Age-structured Forests Subject To Beetle Infestations"

Rachel Tate
6 years ago
Views:

Tree Harvesting In Age-structured Forests Subject To Beetle Infestations Preliminary Report Folashade Agusto Department of Ecology and Evolutionary Biology, University of Kansas Lawrence, KS

1 Tree Harvesting In Age-structured Forests Subject To Beetle Infestations Preliminary Report Folashade Agusto Department of Ecology and Evolutionary Biology, University of Kansas Lawrence, KS The 46th John H. Barrett Memorial Lectures University of Tennessee, Knoxville May 16, 2016 Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To BeetleJ.H. Infestations Barrett / 32

2 Collaborators Maria Leite, Mathematics & Statistics, University of South Florida at St. Petersburg, Benito Chen-Charpentier, Department of Mathematics, University of Texas at Arlington, Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To BeetleJ.H. Infestations Barrett / 32

3 Outline Introduction Forest-Beetle Outbreak Model Forest-Beetle Outbreak and Harvesting Model Summary Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To BeetleJ.H. Infestations Barrett / 32

4 Introduction Disturbance is a major agent in determining the heterogeneity of forest ecosystems. Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To BeetleJ.H. Infestations Barrett / 32

5 Introduction Disturbance is a major agent in determining the heterogeneity of forest ecosystems. Two of the dominant disturbances of North America forests are beetle outbreaks and logging. Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To BeetleJ.H. Infestations Barrett / 32

6 Introduction Disturbance is a major agent in determining the heterogeneity of forest ecosystems. Two of the dominant disturbances of North America forests are beetle outbreaks and logging. According to the National Report on Sustainable Forests The incidence of beetle-induced tree mortality has increased 3-fold in the past decade. Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To BeetleJ.H. Infestations Barrett / 32

Introduction Beetle outbreaks is one of the most disruptive natural disturbances of the North America forests. Natural disturbances: droughts, etc.

7 Introduction Beetle outbreaks is one of the most disruptive natural disturbances of the North America forests. Natural disturbances: droughts, etc. insects and diseases epidemics, wildfire, floods, Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To BeetleJ.H. Infestations Barrett / 32

8 Introduction Logging is an anthropogenic disturbance that has been subject to distinct policies. Anthropogenic disturbances: land-use (includes harvesting ), application of herbicides, etc. Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To BeetleJ.H. Infestations Barrett / 32

9 Introduction Eruptive species such as beetle population are important systems for studying economically and environmentally damaging consequences of anthropogenic activities such as harvesting. Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To BeetleJ.H. Infestations Barrett / 32

10 Introduction Eruptive species such as beetle population are important systems for studying economically and environmentally damaging consequences of anthropogenic activities such as harvesting. Beetle populations can be categorize into Endemic populations and Epidemic populations. Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To BeetleJ.H. Infestations Barrett / 32

11 Introduction Eruptive species such as beetle population are important systems for studying economically and environmentally damaging consequences of anthropogenic activities such as harvesting. Beetle populations can be categorize into Endemic populations and Epidemic populations. Endemic beetle populations play an important role by removing older and less viable trees from the forest. Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To BeetleJ.H. Infestations Barrett / 32

12 Introduction Eruptive species such as beetle population are important systems for studying economically and environmentally damaging consequences of anthropogenic activities such as harvesting. Beetle populations can be categorize into Endemic populations and Epidemic populations. Endemic beetle populations play an important role by removing older and less viable trees from the forest. Epidemic states occur periodically and constitute natural cycles. During the epidemic period forest mortality is significant. Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To BeetleJ.H. Infestations Barrett / 32

13 Introduction Eruptive species such as beetle population are important systems for studying economically and environmentally damaging consequences of anthropogenic activities such as harvesting. Beetle populations can be categorize into Endemic populations and Epidemic populations. Endemic beetle populations play an important role by removing older and less viable trees from the forest. Epidemic states occur periodically and constitute natural cycles. During the epidemic period forest mortality is significant. Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To BeetleJ.H. Infestations Barrett / 32

14 Motivation The long-term goal of forest management is a relatively stable source of wood. Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To BeetleJ.H. Infestations Barrett / 32

15 Motivation The long-term goal of forest management is a relatively stable source of wood. Forest age is a useful variable to analyze the impact of disturbance on tree harvesting benefit. Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To BeetleJ.H. Infestations Barrett / 32

16 Motivation The long-term goal of forest management is a relatively stable source of wood. Forest age is a useful variable to analyze the impact of disturbance on tree harvesting benefit. The goal of many forest management practices is a stable forest age-class structure. Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To BeetleJ.H. Infestations Barrett / 32

17 Motivation The long-term goal of forest management is a relatively stable source of wood. Forest age is a useful variable to analyze the impact of disturbance on tree harvesting benefit. The goal of many forest management practices is a stable forest age-class structure. In the absence of disturbances, this forest management goal could easily be achieved. Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To BeetleJ.H. Infestations Barrett / 32

18 Motivation In the face of recurring beetle outbreaks and other disruptive processes, forest planning and regenerative measures become more difficult. Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To BeetleJ.H. Infestations Barrett / 32

19 Motivation In the face of recurring beetle outbreaks and other disruptive processes, forest planning and regenerative measures become more difficult. Requiring knowledge of the effects on the forest of a wide variety of environmental factors such as Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To BeetleJ.H. Infestations Barrett / 32

20 Motivation In the face of recurring beetle outbreaks and other disruptive processes, forest planning and regenerative measures become more difficult. Requiring knowledge of the effects on the forest of a wide variety of environmental factors such as fire size and frequency, Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To BeetleJ.H. Infestations Barrett / 32

21 Motivation In the face of recurring beetle outbreaks and other disruptive processes, forest planning and regenerative measures become more difficult. Requiring knowledge of the effects on the forest of a wide variety of environmental factors such as fire size and frequency, harvesting, Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To BeetleJ.H. Infestations Barrett / 32

22 Motivation In the face of recurring beetle outbreaks and other disruptive processes, forest planning and regenerative measures become more difficult. Requiring knowledge of the effects on the forest of a wide variety of environmental factors such as fire size and frequency, harvesting, beetle outbreaks, and Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To BeetleJ.H. Infestations Barrett / 32

23 Motivation In the face of recurring beetle outbreaks and other disruptive processes, forest planning and regenerative measures become more difficult. Requiring knowledge of the effects on the forest of a wide variety of environmental factors such as fire size and frequency, harvesting, beetle outbreaks, and drought Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To BeetleJ.H. Infestations Barrett / 32

24 Motivation There has been growing interest in the field in understanding how to manage an age-structured forest in the presence of insect infestation and harvesting. Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To Beetle J.H. Infestations Barrett / 32

25 Motivation There has been growing interest in the field in understanding how to manage an age-structured forest in the presence of insect infestation and harvesting. The individual impact of each of these disturbances on the forest dynamics and its age structure is not completely understood. Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To Beetle J.H. Infestations Barrett / 32

26 Motivation There has been growing interest in the field in understanding how to manage an age-structured forest in the presence of insect infestation and harvesting. The individual impact of each of these disturbances on the forest dynamics and its age structure is not completely understood. The additional complication of the potential interactions between those disturbances as well as the effects on the age profile of the trees population is rarely explored. Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To Beetle J.H. Infestations Barrett / 32

27 Objectives Tree-Beetle Infestation What is the effect of beetle outbreaks on forest trees dynamics? Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To Beetle J.H. Infestations Barrett / 32

28 Objectives Tree-Beetle Infestation What is the effect of beetle outbreaks on forest trees dynamics? Beetle outbreaks on an age-structured forest: ODE-PDE model temporal dynamics (numeric simulations) Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To Beetle J.H. Infestations Barrett / 32

29 Objectives Harvesting Strategies and Maximum Harvesting Benefit What is the effect of beetle outbreaks & harvesting strategies on the harvest benefit? Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To Beetle J.H. Infestations Barrett / 32

Objectives Harvesting Strategies and Maximum Harvesting Benefit What is the effect of beetle outbreaks & harvesting strategies on the harvest benefit?

30 Objectives Harvesting Strategies and Maximum Harvesting Benefit What is the effect of beetle outbreaks & harvesting strategies on the harvest benefit? Harvesting and beetle outbreaks disturbances: ODE-PDE extended model temporal dynamics (numerical simulations) maximum harvesting benefit Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To Beetle J.H. Infestations Barrett / 32

31 Forest-Beetle Outbreaks Model Beetle-ODE System ) db dt = r bb (1 BKe αb2 1 + βb 2 Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To Beetle J.H. Infestations Barrett / 32

32 Forest-Beetle Outbreaks Model Tree-PDE System V (a, t) t + V (a, t) a = [µ(a) + µ B (a)]v (a, t), V (a, 0) = v 0 exp ( (a a 1) 2 ), a [0, A], 2s 2 1 V (0, t) = A 0 b(a)v (a, t)da, t [0, T ]. Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To Beetle J.H. Infestations Barrett / 32

33 Forest-Beetle Outbreaks Model The initial tree age distribution V (a, 0) is described by the function V (a, 0) = v 0 exp ( (a a 1) 2 ), a [0, A]. 2s 2 1 The initial tree age distribution V (a, 0) is a normal distribution Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To Beetle J.H. Infestations Barrett / 32

34 Forest-Beetle Outbreaks Model The birth rate for trees b(a) is described by the function b(a) = b 0 a m (A a)e ca, with m = 2, b 0 > 0. The birth rate b(a) is of lognormal type distribution. Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To Beetle J.H. Infestations Barrett / 32

35 Forest-Beetle Outbreaks Model The death rate for trees µ(a) is described by the function µ(a) = A exp( a A ). (A a) The death rate µ(a) is concave up, strictly increasing, and continuous on [0, a). Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To Beetle J.H. Infestations Barrett / 32

36 Forest-Beetle Outbreaks Model The additional mortality due to the beetles, µ B (a), is assumed to depend on the age of the tree. µ B (a) = f k (a)b, with f k (a) = d exp ( (a a 0) 2 ) 2σ 2, f k (a) gives the age-related distribution of the fraction of trees killed per beetle in a successful attack. The mortality of trees due to beetles µ B (a) is a Gaussian distribution. Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To Beetle J.H. Infestations Barrett / 32

37 Forest-Beetle Outbreaks Model Beetle-Tree ODE-PDE System ) db dt = r bb (1 BKe αb2 1 + βb 2, V (a, t) t + V (a, t) a = [µ(a) + µ B (a)]v (a, t), V (a, 0) = v 0 exp ( (a a 1) 2 ), a [0, A], 2s 2 1 V (0, t) = A 0 b(a)v (a, t)da, t [0, T ]. Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To Beetle J.H. Infestations Barrett / 32

38 Forest-Beetle Outbreaks Model Beetles ODE Dynamics **Chen & Leite (2014)** db dt = r bb (1 BKe ) αb2 1 + βb 2 Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To Beetle J.H. Infestations Barrett / 32

39 Forest-Beetle Outbreaks Model Beetles ODE Dynamics **Chen & Leite (2014)** db dt = r bb (1 BKe ) αb2 1 + βb 2 There is at most four equilibria solutions: B = 0 - trivial equilibrium (unstable) B en - endemic equilibrium (l.a.s.) B ep - epidemic equilibrium (l.a.s.) B T - threshold equilibrium (unstable) Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To Beetle J.H. Infestations Barrett / 32

40 Forest-Beetle Outbreaks Model Solution and Numerical Approach V (a, t) t + V (a, t) a = [µ(a) + µ B (a)]v (a, t), V (a, 0) = v 0 exp ( (a a 1) 2 ), a [0, A], 2s 2 1 V (0, t) = A 0 b(a)v (a, t)da, t [0, T ]. Solution: a function V (a, t) L (0, T ; L 1 (0, A)), absolutely continuous along almost every characteristic line a t = k, with (a, t) [0, A] [0, T ] Existence and uniqueness of the solution was proved by Aniţa, S et al. (2011) Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To Beetle J.H. Infestations Barrett / 32

41 Forest-Beetle Outbreaks Model Solution and Numerical Approach removal of the singularity (Pelovska & Iannelli (2006)) integrate numerically the B- ODE results are then fed into an implicit finite difference scheme Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To Beetle J.H. Infestations Barrett / 32

42 Forest-Beetle Outbreaks Model Solution and Numerical Approach removal of the singularity (Pelovska & Iannelli (2006)) integrate numerically the B- ODE results are then fed into an implicit finite difference scheme Consider 3 scenarios: no beetles; beetles endemic state; beetles epidemic state Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To Beetle J.H. Infestations Barrett / 32

43 Forest-Beetle Outbreaks Model Trees & Beetles Dynamics: Numerical Simulations Left panels: Beetles endemic state Right panels: Beetles epidemic state Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To Beetle J.H. Infestations Barrett / 32

44 Forest-Beetle Outbreaks Model Trees Mortality Due to Beetles Left panel: Beetles endemic state Right panel: Beetles epidemic state Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To Beetle J.H. Infestations Barrett / 32

45 Forest-Beetle Outbreaks Model Trees Age Distribution Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To Beetle J.H. Infestations Barrett / 32

46 Forest-Beetle Outbreak and Harvesting Model Mathematical Model Incorporating Harvesting ) db dt = r bb (1 BKe αb2 1 + βb 2, V (a, t) t + V (a, t) a = [µ(a) + µ B (a)]v (a, t) H(a, V), V (a, 0) = v 0 exp ( (a a 1) 2 ) 2s 2 1 a [0, A], V (0, t) = A 0 b(a)v (a, t)da, t [0, T ]. Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To Beetle J.H. Infestations Barrett / 32

47 Forest-Beetle Outbreak and Harvesting Model Harvesting Scenarios Harvesting step-function 0 0 a < a H(a, V) = a a h max h max - maximum number of trees of an age a a present and a - cut age. Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To Beetle J.H. Infestations Barrett / 32

48 Forest-Beetle Outbreak and Harvesting Model Harvesting Scenarios Harvesting step-function 0 0 a < a H(a, V) = a a h max h max - maximum number of trees of an age a a present and a - cut age. Constant rate harvesting { 0 0 a < a H(a, V) = h V (a, t) a a h - fixed proportion of trees of age a a that are cut. Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To Beetle J.H. Infestations Barrett / 32

49 Forest-Beetle Outbreak and Harvesting Model Harvest Benefit & Numerical Simulations What is the effect of beetle outbreaks & harvesting strategies on the harvest benefit? T A 0 0 V (a, t) da dt Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To Beetle J.H. Infestations Barrett / 32

50 Forest-Beetle Outbreak and Harvesting Model Harvest Benefit & Numerical Simulations What is the effect of beetle outbreaks & harvesting strategies on the harvest benefit? T A 0 0 V (a, t) da dt Consider 3 scenarios: no beetles; beetles endemic state; beetles epidemic state Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To Beetle J.H. Infestations Barrett / 32

51 Forest-Beetle Outbreak and Harvesting Model Harvest Benefit & Numerical Simulations Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To Beetle J.H. Infestations Barrett / 32

52 Forest-Beetle Outbreak and Harvesting Model Harvest Benefit & Numerical Simulations Harvesting step-function Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To Beetle J.H. Infestations Barrett / 32

53 Forest-Beetle Outbreak and Harvesting Model Harvest Benefit & Numerical Simulations Harvesting step-function Constant rate harvesting Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To Beetle J.H. Infestations Barrett / 32

54 Summary Summary Introduce fairly general deterministic mathematical models to explore the effect of harvesting and beetle outbreak disturbances interaction on the forest dynamics. Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To Beetle J.H. Infestations Barrett / 32

55 Summary Summary Introduce fairly general deterministic mathematical models to explore the effect of harvesting and beetle outbreak disturbances interaction on the forest dynamics. As expected the total numbers of trees killed during beetle epidemic state is significantly higher than during the beetle endemic state. Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To Beetle J.H. Infestations Barrett / 32

56 Summary Summary Introduce fairly general deterministic mathematical models to explore the effect of harvesting and beetle outbreak disturbances interaction on the forest dynamics. As expected the total numbers of trees killed during beetle epidemic state is significantly higher than during the beetle endemic state. If wood source is the only benefit considered, the harvesting step-function is preferred over constant rate strategy. Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To Beetle J.H. Infestations Barrett / 32

57 Summary Summary Introduce fairly general deterministic mathematical models to explore the effect of harvesting and beetle outbreak disturbances interaction on the forest dynamics. As expected the total numbers of trees killed during beetle epidemic state is significantly higher than during the beetle endemic state. If wood source is the only benefit considered, the harvesting step-function is preferred over constant rate strategy. Future Directions Parameterize the model using data. Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To Beetle J.H. Infestations Barrett / 32

58 Summary Summary Introduce fairly general deterministic mathematical models to explore the effect of harvesting and beetle outbreak disturbances interaction on the forest dynamics. As expected the total numbers of trees killed during beetle epidemic state is significantly higher than during the beetle endemic state. If wood source is the only benefit considered, the harvesting step-function is preferred over constant rate strategy. Future Directions Parameterize the model using data. Study an optimal control problem to minimize disturbances and maximize profit. Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To Beetle J.H. Infestations Barrett / 32

59 Summary Summary Introduce fairly general deterministic mathematical models to explore the effect of harvesting and beetle outbreak disturbances interaction on the forest dynamics. As expected the total numbers of trees killed during beetle epidemic state is significantly higher than during the beetle endemic state. If wood source is the only benefit considered, the harvesting step-function is preferred over constant rate strategy. Future Directions Parameterize the model using data. Study an optimal control problem to minimize disturbances and maximize profit. Build models including other disturbances (e.g., wildfire, drought) Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To Beetle J.H. Infestations Barrett / 32

60 Now Recruiting Grad Students Immuno-Epidemiological Model for Malaria Multi-Scale Model for Multi-Host Single Pathogen System Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To Beetle J.H. Infestations Barrett / 32

61 Now Recruiting Grad Students Immuno-Epidemiological Model for Malaria Multi-Scale Model for Multi-Host Single Pathogen System Thank you. Questions!!! Folashade Agusto (KU) Tree Harvesting In Age-structured Forests Subject To Beetle J.H. Infestations Barrett / 32

Synergies, feedbacks and tipping points: mountain pine beetle s rapid range expansion threatens invasion of North American boreal pine forests

Synergies, feedbacks and tipping points: mountain pine beetle s rapid range expansion threatens invasion of North American boreal pine forests Allan L. Carroll The University of British Columbia Department