Philippa Kaur

The long-term population dynamics (i.e., extinction or persistence in an equilibrium value) of the system (2) is mainly dependent on the constant allocation of energy between life-history functions. In the case of population persistence, there is a unique long-term allocation strategy towards reproductive functions that maximizes the population abundance and minimizes individual consumption. In addition, this strategy is dependent on the parameters associated with both fertility ((kappa ) and ({mathscr {B}}_0)) and survival ((xi _{c}) and (e_{1/2})) costs.
Preliminary results
To investigate long-term dynamic patterns, we will relate the energy states of reproduction and maintenance, resource density and population abundance at the end of reproductive seasons, namely in the time sequence ({ntau ^+}_{nge 0}). In the system (4) the relationships between the state variables are the same for each (tau ) unit of time. Then, there is a transformation that relates to the vector ( (R,E_ {r}, E_ {m},P) ((n + 1)tau ^+) ) with ( (R,E_ {r}, E_ {m}, P) (n tau ^+) ) such that ( (n + 1) tau ^+ – ntau ^+ = tau ). This relationship is determined by the following discretization (or stroboscopic map) of the impulsive differential system (4):

$$begin{aligned} left{ begin{array}{lll} E_{r}((n+1)tau ^+)&=(1-kappa gamma )[E_{r}(ntau ^+)+alpha Phi (n,P(ntau ^+))],\ &{}&{}\ E_{m}((n+1)tau ^+)&=dfrac{{mathscr {B}}_0[E_{m}(ntau ^+)+(1-alpha )xi _{c}Phi (n,P(ntau ^+))]}{{mathscr {B}}_0+gamma [E_{r}(ntau ^+)+alpha Phi (n,P(ntau ^+))]},\ P((n+1)tau ^+)&=left{ 1-mu +dfrac{gamma [E_{r}(ntau ^+)+alpha Phi (n,P(ntau ^+))]E_{m}((n+1)tau ^+)}{e_{1/2}+E_{m}((n+1)tau ^+)}right} P(ntau ^+)e^{-lambda tau } end{array} right. end{aligned}$$
(6)

and (R(ntau ^+)=K), where the function (Phi ) defined by (5) is evaluated at (t=(n+1)tau ) and extended to (p=0):

$$begin{aligned} Phi (n,p)=left{ begin{array}{ll} dfrac{R_{max}(e^{lambda tau }-1)}{lambda }, &quad text {if}, p=0, \ &{}\ dfrac{K-R((n+1)tau )e^{lambda tau }}{p}+dfrac{lambda }{p}displaystyle int _{ntau }^{(n+1)tau }{R(s)e^{lambda (s-ntau )}ds}, & quad text {if}, pne 0, end{array} right. end{aligned}$$
(7)

for any (nge 0). Indeed, in non-reproductive seasons the consumer–resource dynamics are described by the continuous component of the system (4), which can be solved. Directly we have (P(t)=P(ntau ^+)e^{-lambda (t-ntau )}) for any (tin (ntau ,(n+1)tau ]). In addition, the per capita consumption rate (1) can be expressed in terms of ( R'(t)). Therefore, reproductive and maintenance energy rates are determined by

$$begin{aligned} E_r'(t)=-alpha frac{R'(t)e^{lambda (t-ntau )}}{P(ntau ^+)}quad text {and}quad E_m'(t)=-(1-alpha )xi _{c}frac{R'(t)e^{lambda (t-ntau )}}{P(ntau ^+)},quad tin (ntau ,(n+1)tau ]. end{aligned}$$

Integrating these functions in the interval ( (n tau , t] ), we obtain

$$begin{aligned} E_r(t)&= {} E_{r}(ntau ^+)+alpha Phi (t,R(t),P(ntau ^+)) end{aligned}$$
(8)

$$begin{aligned} E_m(t)&= {} E_m(ntau ^+)+(1-alpha )xi _{c}Phi (t,R(t),P(ntau ^+)). end{aligned}$$
(9)

Evaluating Eqs. (8)–(9) at ( t = (n + 1)tau ) (end of the non-reproductive season), we have

$$begin{aligned} E_r((n+1)tau )&= {} E_{r}(ntau ^+)+alpha Phi (n,P(ntau ^+)), end{aligned}$$
(10)

$$begin{aligned} E_m(n+1)tau )&= {} E_m(ntau ^+)+(1-alpha )xi _{c}Phi (n,P(ntau ^+)), end{aligned}$$
(11)

with (Phi (n,P(ntau ^+))= Phi ((n+1)tau ,R((n+1)tau ),P(ntau ^+))) and (P((n+1)tau )=P(ntau ^+)e^{-lambda tau }) for any (nge 0). In addition, evaluating the discrete component of system (4) at (t=(n+1)tau ), we obtain

$$begin{aligned} E_{r}((n+1)tau ^+)&= {} (1-kappa gamma )E_{r}((n+1)tau ), end{aligned}$$
(12)

$$begin{aligned} E_m((n+1)tau ^+)&= {} frac{{mathscr {B}}_0E_m((n+1)tau )}{{mathscr {B}}_0+gamma E_r((n+1)tau )}, end{aligned}$$
(13)

$$begin{aligned} P((n+1)tau ^+)= & {} left[ 1-mu +gamma E_{r}((n+1)tau )frac{E_{m}((n+1)tau ^+)}{e_{1/2}+E_{m}((n+1)tau ^+)}right] P((n+1)tau ). end{aligned}$$
(14)

Finally, substituting the Eqs. (10)–(11) into the Esq. (12)–(14) we obtain the discretization given by Eq. (6).
In order to obtain the equilibrium points of system (6), we can solve the following equations:

$$begin{aligned} e_{r}&= {} (1-kappa gamma )[e_{r}+alpha Phi (rho )],\ e_{m}&= {} dfrac{{mathscr {B}}_{0}[e_{m}+(1-alpha )xi _{c}Phi (rho )]}{{mathscr {B}}_0+gamma [e_{r}+alpha Phi (rho )]},\ rho&= {} left{ 1-mu +gamma [e_{r}+alpha Phi (rho )]dfrac{e_{m}}{e_{1/2}+e_{m}}right} rho e^{-lambda tau }, end{aligned}$$

where (e_{r}:=lim _{nrightarrow +infty }{E_{r}(ntau ^+)}), (e_{m}:=lim _{nrightarrow +infty }{E_{m}(ntau ^+)}), (rho :=lim _{nrightarrow +infty }{P(ntau ^+)}) and then, (Phi (rho ):=lim _{nrightarrow +infty }{Phi (n,P(ntau ^+))}). From the first equation, we have (e_{r}=(1-kappa gamma )alpha Phi (P)/kappa gamma ) and then (e_{m}=kappa {mathscr {B}}_0 (1-alpha )xi _{c}/alpha ) for (Pin {0,rho }) such that (Phi (0)=R_{max}(e^{lambda tau }-1)/lambda ) and

$$begin{aligned} Phi (rho )={mathscr {A}},quad text {where}quad {mathscr {A}}=dfrac{e^{lambda tau }-1+mu }{{mathscr {B}}_0}cdot left{ dfrac{e_{1/2}xi _{c}^{-1}}{1-alpha } +dfrac{kappa {mathscr {B}}_0}{alpha }right} . end{aligned}$$
(15)

Therefore, assuming (lambda =0), we have (Gamma (rho )=K-{mathscr {A}}rho ) (from equations (7) and (15)) where (Gamma :=lim _{nrightarrow +infty }{R((n+1)tau )}), (rho ) is the solution of

$$begin{aligned} r_0ln left( dfrac{K-{mathscr {A}}rho }{K}right) -{mathscr {A}}=-R_{max}tau , end{aligned}$$

if, and only if,

$$begin{aligned} rho =dfrac{K}{{mathscr {A}}}left[ 1-exp left( dfrac{{mathscr {A}}-R_{max}tau }{r_0}right) right] . end{aligned}$$
(16)

Long-term population dynamics
From the discretization (6), there are two dynamic behaviors for the long-term population abundance: extinction (see Fig. 2a) and persistence (see Fig. 2b).
Figure 2

Long-term behavior of solutions of the model (4). Peak values correspond to the solution of the discrete model (6) in its population component. (a) Extinction behavior, considering (alpha in {0.1,,0.8}) as energy allocation strategy toward reproductive and (b) persistence behavior considering (alpha in {0.35,, 0.55}). We consider the following parameter set (eta =(2,0.5,2,0.25,500,2,1,0.5,0.5,0.86,alpha ,0.1)) where the constant of fertility costs is described by (kappa =(1+gamma )^{-1}).

Full size image

The differentiation of these behaviors strongly depends on the individual consumption of resources defined throughout each non-reproductive season by

$$begin{aligned} C_{I}(t) = frac{[K-R(t)]e^{lambda (t-ntau )}}{P(ntau ^+)},, tin (ntau ,(n+1)tau ]. end{aligned}$$

At the end of each non-reproductive season, namely at the time ( t = (n + 1) tau ), the individual consumption is given by

$$begin{aligned} C_I((n+1)tau ) = frac{[K-R((n+1)tau )]e^{lambda tau }}{P(ntau ^+)}, end{aligned}$$
(17)

where (R((n+1)tau )) is the non-consumed resource density by the population, the amount that is obtained from the implicit solution of the resource density equation in the continuous component of the system (4). Thus, projecting the individual consumption of the resource into the long term, and taking (Gamma (rho )=(K-{mathscr {A}}rho +lambda {mathscr {I}})e^{-lambda tau }), the expression (17) assumes the form

$$begin{aligned} C_{I}^{infty }(rho )=left{ begin{array}{ll} {mathscr {A}}+dfrac{K(e^{lambda tau }-1)-lambda {mathscr {I}}}{rho },&{}quad text {if}, rho ne 0,\ &{}\ dfrac{R_{max}(e^{lambda tau }-1)}{lambda },&{}quad text {if}, rho =0, end{array}right. end{aligned}$$
(18)

where ({mathscr {I}}:=lim _{nrightarrow +infty }{int _{ntau }^{(n+1)tau }{R(s)e^{lambda (s-ntau )}ds}}). On the one hand, the individual consumption ( C_{I}^{infty } ) is composed of a basis amount corresponding to the term ({mathscr {A}} ) and an amount resulting from the equal division of a resource not consumed by individuals dying during the non-reproductive season, ( K(e^{lambda tau } -1) – lambda {mathscr {I}}). Furthermore, when the population experiences a reduced mortality during the non-reproductive season (i.e., ( lambda approx 0) ), the individual consumption is ( C_{I}^{infty } (rho ) approx {mathscr {A}} ). On the other hand, whether the population abundance is low, the per capita resource is high, implying an individual consumption close to (R_ {max} (e^{lambda tau } -1) / lambda ) (equivalent to taking the limit of ( C_{I}((n+1)tau )) as ( P(ntau ^+)rightarrow 0 )). Certainly, this quantity does not represent the effective individual consumption, but rather establishes an upper limit for this and therefore represents a value of non-persistence. Thus, behaviors related to the equilibrium solutions of the discrete system (6) can be differentiated by the threshold value

$$begin{aligned} {mathscr {U}}=dfrac{C_{I}^{infty }(0)}{C_{I}^{infty }(rho )}. end{aligned}$$

In particular, when the mortality of the population is low, the long-term abundance is described by Eq. (16) and the threshold value assumes the following form ({mathscr {U}}=R_{max}tau /{mathscr {A}}). Thus, we conclude that the persistence of the population is established when ({mathscr {U}} >1) and extinction when ({mathscr {U}}le 1).
Finally, we can see that the stabilization of population size in the long term is in response to a dense-dependent behaviour where the per capita growth rate in the long term is (r_{infty }:={alpha {mathscr {A}}/kappa }{mathscr {S}}(e_m)) equivalent to mortality fraction (mu ), where (e_{m}=kappa {mathscr {B}}_{0}(1-alpha )xi _{c}/alpha ) is the equilibrium value of energy maintenance after the reproductive season. In addition, the derivative of r with respect to P is

$$begin{aligned} dfrac{dr}{dP} = gamma dfrac{dPhi }{d P}left{ alpha {mathscr {S}}(E_{m})+(E_{r}+alpha Phi )dfrac{d{mathscr {S}}}{dE_{m}}cdot dfrac{dE_{m}}{dPhi }right} , end{aligned}$$

where (dPhi /d P1) then the long-term population behavior is persistence.

Proof
We divide the proof into two cases: (kappa gamma =1) and (0 More

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To produce our global Forest Landscape Integrity Index (FLII), we combined four sets of spatially explicit datasets representing: (i) forest extent23; (ii) observed pressure from high impact, localized human activities for which spatial datasets exist, specifically: infrastructure, agriculture, and recent deforestation27; (iii) inferred pressure associated with edge effects27, and other diffuse processes, (e.g., activities such as hunting and selective logging)27 modeled using proximity to observed pressures; and iv) anthropogenic changes in forest connectivity due to forest loss27 (see Supplementary Table 1 for data sources). These datasets were combined to produce an index score for each forest pixel (300 m), with the highest scores reflecting the highest forest integrity (Fig. 1), and applied to forest extent for the start of 2019. We use globally consistent parameters for all elements (i.e., parameters do not vary geographically). All calculations were conducted in Google Earth Engine (GEE)60.
Forest extent
We derived a global forest extent map for 2019 by subtracting from the Global Tree Cover product for 200023 annual Tree Cover Loss 2001–2018, except for losses categorized by Curtis and colleagues24 as those likely to be temporary in nature (i.e., those due to fire, shifting cultivation and rotational forestry). We applied a canopy threshold of 20% based on related studies e.g.31,61, and resampled to 300 m resolution and used this resolution as the basis for the rest of the analysis (see Supplementary Note 1 for further methods).
Observed human pressures
We quantify observed human pressures (P) within a pixel as the weighted sum of impact of infrastructure (I; representing the combined effect of 41 types of infrastructure weighted by their estimated general relative impact on forests (Supplementary Table 3), agriculture (A) weighted by crop intensity (indicated by irrigation levels), and recent deforestation over the past 18 years (H; excluding deforestation from fire, see Discussion). Specifically, for pixel i:

$${mathrm{P}}_{mathrm{i}} = {mathrm{exp}}left( { – {upbeta}_1{mathrm{I}}_{mathrm{i}}} right) + {mathrm{exp}}left( { – {upbeta}_2{mathrm{A}}_{mathrm{i}}} right) + {mathrm{exp}}left( { – {upbeta}_3{mathrm{H}}_{mathrm{i}}} right)$$
(1)

whereby the values of β were selected so that the median of the non-zero values for each component was 0.75. This use of exponents is a way of scaling variables with non-commensurate units so that they can be combined numerically, while also ensuring that the measure of observed pressure is sensitive to change (increase or decrease) in the magnitude of any of the three components, even at large values of I, A, or H. This is an adaptation of the Human Footprint methodology62. See Supplementary Note 3 for further details.
Inferred human pressures
Inferred pressures are the diffuse effects of a set of processes for which directly observed datasets do not exist, that include microclimate and species interactions relating to the creation of forest edges63 and a variety of intermittent or transient anthropogenic pressures such as selective logging, fuelwood collection, hunting; spread of fires and invasive species, pollution, and livestock grazing64,65,66. We modeled the collective, cumulative impacts of these inferred effects through their spatial association with observed human pressure in nearby pixels, including a decline in effect intensity according to distance, and partitioning into stronger short-range and weaker long-range effects. The inferred pressure (P′) on pixel i from source pixel j is:

$$Pprime _{i,j} = P_jleft( {w_{i,j} + v_{i,j}} right)$$
(2)

where wi,j is the weighting given to the modification arising from short-range pressure, as a function of distance from the source pixel, and vi,j is the weighting given to the modification arising from long-range pressures.
Short-range effects include most of the processes listed above, which together potentially affect most biophysical features of a forest, and predominate over shorter distances. In our model, they decline exponentially, approach zero at 3 km, and are truncated to zero at 5 km (see Supplementary Note 4).

$$begin{array}{l}{mathrm{w}}_{i,j} = alpha ,{mathrm{exp}}( – lambda {mathrm{d}}_{i,j}),,,,,,[{mathrm{for}},{mathrm{d}}_{{mathrm{i,j}}} le {mathrm{5km}}]\ {mathrm{w}}_{i,j} = {mathrm{0}},,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,[{mathrm{for}},{mathrm{d}}_{i,j} > {mathrm{5km}}]end{array}$$
(3)

where α is a constant set to ensure that the sum of the weights across all pixels in the range is 1.85 (see below), λ is a decay constant set to a value of 1 (see67 and other references in Supplementary Note 4) and di,j is the Euclidean distance between the centers of pixels i and j expressed in units of km.
Long-range effects include over-exploitation of high socio-economic value animals and plants, changes to migration and ranging patterns, and scattered fire and pollution events. We modeled long-range effects at a uniform level at all distances below 6 km and they then decline linearly with distance, conservatively reaching zero at a radius of 12 km65,68 (and other references in Supplementary Note 4):

$$begin{array}{l}{mathrm{v}}_{i,j} = gamma ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,[for,d_{i,j} le 6km]\ {mathrm{v}}_{i,j} = gamma left( {12 – d_{i,j}} right)/6,,,,[{mathrm{for}},6{mathrm{km}}, < ,{mathrm{d}}_{i,j} le 12{mathrm{km}}]\ {mathrm{v}}_{i,j} = 0,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,[for,{mathrm{d}}_{i,j} > 12{mathrm{km}}]end{array}$$
(4)

where γ is a constant set to ensure that the sum of the weights across all pixels in the range is 0.15 and di,j is the Euclidean distance between the centers of pixels i and j, expressed in kilometers.
The form of the weighting functions for short- and long-range effects and the sum of the weights (α + γ) were specified based on a hypothetical reference scenario where a straight forest edge is adjacent to a large area with uniform human pressure, and ensuring that in this case total inferred pressure immediately inside the forest edge is equal to the pressure immediately outside, before declining with distance. γ is set to 0.15 to ensure that the long-range effects conservatively contribute no more than 5% to the final index in the same scenario, based on expert opinion and supported e.g., Berzaghi et al.69 regarding the approximate level of impact on values that would be affected by severe defaunation and other long-range effects.
The aggregate effect from inferred pressures (Q) on pixel i from all n pixels within range (j = 1 to j = n) is then the sum of these individual, normalized, distance-weighted pressures, i.e.,

$$Q_i = mathop {sum}_{j=1}^{n} {P{prime}_{i,j}}$$
(5)

Loss of forest connectivity
Average connectivity of forest around a pixel was quantified using a method adapted from Beyer et al.70. The connectivity Ci around pixel i surrounded by n other pixels within the maximum radius (numbered j = 1, 2…n) is given by:

$${mathrm{C}}_i = mathop {sum}_{j=1}^{n} {left( {{mathrm{F}}_j{mathrm{G}}_{i,j}} right)}$$
(6)

where Fj is the forest extent is a binary variable indicating if forested (1) or not (0) and Gi,j is the weight assigned to the distance between pixels i and j. Gi,j uses a normalized Gaussian curve, with σ = 20 km and distribution truncated to zero at 4σ for computational convenience (see Supplementary Note 2). The large value of σ captures landscape connectivity patterns operating at a broader scale than processes captured by other data layers. Ci ranges from 0 to 1 (Ci∈[0,1]).
Current Configuration (CCi) of forest extent in pixel i was calculated using the final forest extent map and compared to the Potential Configuration (PC) of forest extent without extensive human modification, so that areas with naturally low connectivity, e.g., coasts and natural vegetation mosaics, are not penalized. PC was calculated from a modified version of the map of Laestadius et al38. and resampled to 300 m resolution (see Supplementary Note 2 for details). Using these two measures, we calculated Lost Forest Configuration (LFC) for every pixel as:

$${mathrm{LFC}}_i = 1 – left( {{mathrm{CC}}_i/{mathrm{PC}}_i} right)$$
(7)

Values of CCi/PCi  > 1 are assigned a value of 1 to ensure that LFC is not sensitive to apparent increases in forest connectivity due to inaccuracy in estimated potential forest extent – low values represent least loss, high values greatest loss (LFCi∈[0,1]).
Calculating the Forest Landscape Integrity Index
The three constituent metrics, LFC, P, and Q, all represent increasingly modified conditions the larger their values become. To calculate a forest integrity index in which larger values represent less degraded conditions we, therefore, subtract the sum of those components from a fixed large value (here, 3). Three was selected as our assessment indicates that values of LFC + P + Q of 3 or more correspond to the most severely degraded areas. The metric is also rescaled to a convenient scale (0-10) by multiplying by an arbitrary constant (10/3). The FLII for forest pixel i is thus calculated as:

$${mathrm{FLII}}_i = left[ {10/3} right] (3 – {mathrm{min}}(3,,[P_i + Q_i + {mathrm{LFC}}_i]))$$
(8)

where FLIIi ranges from 0 to 10, forest areas with no modification detectable using our methods scoring 10 and those with the most scoring 0.
Illustrative forest integrity classes
Whilst a key strength of the index is its continuous nature, the results can also be categorized for a range of purposes. In this paper three illustrative classes were defined, mapped, and summarized to give an overview of broad patterns of integrity in the world’s forests. The three categories were defined as follows.
High Forest Integrity (scores ≥ 9.6) Interiors and natural edges of more or less unmodified naturally regenerated (i.e., non-planted) forest ecosystems, comprised entirely or almost entirely of native species, occurring over large areas either as continuous blocks or natural mosaics with non-forest vegetation; typically little human use other than low-intensity recreation or spiritual uses and/or low-intensity extraction of plant and animal products and/or very sparse presence of infrastructure; key ecosystem functions such as carbon storage, biodiversity, and watershed protection and resilience expected to be very close to natural levels (excluding any effects from climate change) although some declines possible in the most sensitive elements (e.g., some high value hunted species).
Medium Forest Integrity (scores  > 6.0 but More

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