SUPPLEMENTAL I. MATHEMATICAL ANALYSES.
1. General Model - Part I
In the main text we present and explain the motivation behind the resource-based model we study here. The
dynamical states are simply a set of differential equations in the variables human population biomass φ and
average niche state ω. The system looks as
(
φ0 = f ω − m φ(1 − φ),
(1)
ω 0 = λ(1 − ω) − (B + Ef ω)φ,
Fixed points are found by solving the previous set of equations in the time-independent case. Since equations
are non-linear, several solutions are found that correspond to the different scenarios presented in the main
text:
• Extinction: corresponding to the case φ = 0 implying that niche factor state ω is necessarily 1. By
performing a linear stability analysis, the eigenvalues of the Jacobian matrix are given by f − m and
−λ. Necessarily, the stability of the extinction scenario is related to the condition
f < m,
otherwise, this scenario loses its stability and extinction is no longer observed.
• Niche Limitation: Here, we look for a solution to the system such that variable ω is different from
1. More precisely, we impose that the population size is limited by the average niche state at large
times. If f > m, then for the existence of such a steady state we have ω = m/f and
λ(1 − m/f )
.
B + Em
• Habitat Limitation: The last equilibria is related to the value φ = 1, meaning average niche state is
sufficient and human biomass ”fills” its habitat. From the second equation it follows that ω is given
by
λ−B
ω = H(λ) :=
.
λ + Ef
For the linear stability analysis, the Jacobian eigenvalues are −(λ + Ef ) and −(f H(λ) − m), and it
follows that the niche state domination equilibria is asymptotically stable if
φ=
H(λ) > m/f.
Finally, noticing that H 0 (λ) > 0 then there exists λ∗ such that if f > m, the following dichotomy
holds
– For any value λ > λ∗ the Habitat Limitation equilibrium is stable;
– As soon as λ becomes smaller that λ∗ the Habitat Limitation equilibrium looses its stability and
the Niche Limitation equilibrium kicks in.
2. Part II: Adjusting the general model to human population
The general model is transformed in order to include the technological effects upon the rates of improvements
and impacts on the average niche state variable, as well as keeping track on the technological stock µ. In
particular, the following set of equations are proposed:
(2)

0

φ = f ω − m φ(1 − φ),
ω 0 = c1 µα (1 − ω) − (c2 µβ + Ef ω)φ,

 0
µ = (g + ρµ)φ − l(µ − ε)
The set of fixed points is more complicated than the previous general model, and stability conditions can not
be trivially listed.
On the other hand, linear stability of each fixed point (φ, ω, µ) is directly related to the larger eigenvalue of
the matrix


(f ω − m)(1 − 2φ)
f φ(1 − φ)
0
J(φ, ω, µ) :=  −(c2 µβ + Ef ω) −(c1 µα + Ef φ) αc1 µα−1 (1 − ω) − βc2 µβ−1 φ .
g + ρµ
0
ρφ − l
1
2
Indeed, if the larger eigenvalue is strictly negative, there is linear stability and nonlinear exponential convergence for initial conditions inside the attraction domain.
We can group the possibles long-term outcomes in three categories:
• Extinction: by taking φ = 0 it is straightforward that the the only solution to the system is given by
ω = 1 and µ = ε. Jacobian matrix becomes


f −m
0
0
J(0, 1, ε) := −(c2 εβ + Ef ) −c1 εα 0 
g + ρε
0
−l
since the matrix is upper triangular, eigenvalues are (f − m), −c1 εα and −l. In conclusion, as soon
as mortality is larger than fertility f − m < 0 extinction scenario is stable.
• Habitat Limitation: this scenario corresponds to the case of population size reaching the carrying
capacity φ = 1. The other two variables are given by
g + lε
c1 µα − c2 µβ
,
ω=
.
l−ρ
c1 µα + Ef
Defining θ = α − β and c0 = c1 /c2 , the average niche state fixed point can be re-written by
µ=
c0 µθ − 1
c1 µα − c2 µβ
=
,
α
θ
−β
c1 µ + Ef
c0 µ + Ef c−1
2 µ
and by simplicity we assume that the contribution Ef is negligible, getting
ω=
−θ
ω = 1 − c−1
.
0 µ
Jacobian matrix becomes

−(f ω − m)
0
−c1 µα
J(1, ω, µ) :=  −c2 µβ
g + ρµ
0

0
αc1 µα−1 (1 − ω) − βc2 µβ−1  ,
ρ−l
and the eigenvalues are
−(f ω − m),
−c1 µα ,
ρ − l.
From here we get two stability conditions: first it is necessary that ρ < l meaning that the per capita
innovation rate is smaller than the cultural loss rate. This condition also ensures that the fixed point
µ is positive therefore the system is self-consistent. Second, we get that necessarily
m
−θ
1 − c−1
> .
0 µ
f
This second condition is very interesting and plays a fundamental role in the system for large values1
of µ. In particular
– Competent technological efficiency: assume that α−β = θ > 0 and that the fixed point µ is large
enough (it can be obtained either by taking g large or l close to ρ). In that case, the left-hand
side of the previous inequality goes to 1, therefore stability condition holds. We conclude that in
this case the fixed point is stable and nonlinear system converges to the niche state domination
scenario.
– Incompetent technological efficiency: assume now that α − β = θ < 0 and once again that the
fixed point µ is large enough. In that case, at some point stability condition is not true anymore.
In this particular case, we cannot get more precise predictions because the large time behavior
of the solutions is closely related to the initial value of the system. Nonetheless, we do notice
−θ
that as µ increases, the fixed value ω = 1 − c−1
decreases to 0 and therefore, any small
0 µ
perturbation of φ = 1 moves away this value quickly.
– Sufficient technological efficiency: finally, assume that α−β = θ = 0, stability condition becomes
simply
m
.
1 − c−1
0 >
f
if re-scaling parameter c0 is such that previous condition holds, then fixed point is stable and µ
converges to the fixed point.
The conclusion we draw from here is that in the case of ρ < l, at large times it is not possible to have
arbitrarily large values of µ.
1Those values can be obtained by using a large value of g or assuming that ρ and l are close enough.
3
• Niche limitation: in this scenario average niche state takes value ω = ω ∗ := m/f , therefore restricting
the possible population size φ and the technology µ. According to the value of θ different sub-scenarios
are possible. Solutions to the steady values of φ and µ cannot be trivially listed and we only present
the following parametric formulas2
l(µ − ε)
c1 µα (1 − ω ∗ )
gφ + lε
=
φ=
,
µ=
.
c2 µβ + Ef ω ∗
g + ρµ
l − ρφ
Using variables c0 and θ and assuming that E is small with respect to the values of µ, we get
gφ + lε
φ = c0 µθ (1 − ω ∗ ),
µ=
.
l − ρφ
Notice that function l(µ − ε)/(g + ρµ) is strictly increasing from −lε/g to l/ρ as µ increases. On the
other hand function h(µ) := c0 µθ (1 − ω ∗ ) has different shapes depending on the value θ, in particular
– Competent technological efficiency: if α − β = θ > 0 then h(µ) is a strictly increasing function
that starts from 0 and diverges to infinite. Depending on the values of the parameters we can
have either none, one or two solutions of φ, all of them are positive and smaller than l/ρ.
– Incompetent technological efficiency: if α − β = θ < 0 then h(µ) is a strictly decreasing function
that starts from +∞ and converges to 0. We conclude that for any set of parameters there is
only one φ solving the equation, it is positive and smaller than l/ρ.
– Sufficient technological efficiency: finally if α − β = θ = 0 then function h(µ) is constant and
equal to c0 (1 − ω ∗ ). A unique solution φ exists if and only if c0 < l/ρ(1 − ω ∗ ).
Finally, if l − ρφ < 0 then there is no fixed point for µ such that ω ∗ = m/f .
2.1. Infinite technology analysis. Once again we have to distinguish between the niche limitation and
the habitat limitation equilibrium cases. To have a notion of the stability of the system we assume first that
µ is a parameter and then take the limit as µ goes to infinite. System is reduced to
(
φ0 = f ω − m φ(1 − φ),
(3)
ω 0 = c1 µα (1 − ω) − c2 µβ φ,
• Habitat Limitation: recall that the steady state is φ = 1 and
−θ
ω = 1 − c−1
0 µ
then
– Competent technological efficiency: if α − β = θ > 0 then average niche state converges to 1 and
the carrying capacity φ = 1 is stable.
– Incompetent technological efficiency: if on the contrary α − β = θ < 0 then niche state goes to
0 and φ = 1 becomes unstable.
– Suffcient technological efficiency: in this last case, if α − β = θ = 0 then once again carrying
capacity φ = 1 is stable if and only if 1 − c−1
0 > m/f.
• Niche Limitation: now we have ω = m/f and
φ = c0 µθ (1 − ω ∗ )
and population can go to the carrying capacity if and only if θ = α − β > 0, more precisely, it
is necessary to have technological development scaling larger than technological costs scaling, or
technology needs to be competent. However, this fixed point seems to be unstable since at the limit
µ = ∞ the dynamics on the variable ω becomes singular.
2Notice that existence and uniqueness of solutions to this nonlinear algebraic equation is not entirely obvious.